And what situations should I try and avoid in regards to wildlife and areas prone to hazards in the UK?
Edited my question to cover the other answers more as they were very good additions to the wildlife aspect.
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In the UK we have, comparative to other countries, very few venomous creatures. However the false widow and adders are still a risk.
False widows have mostly been recorded in the South of England and Wales (though I've personally never seen one) but have been recorded to have bitten and hospitalise people, whilst this maybe when out and about it has also occurred when people are asleep in their own homes. There are other spiders that are capable of biting humans, but of the 650 or so species in the UK only 15 or so can do so. False widows are also considered non aggressive.
Adders are the only venomous snake in Britain (and again after 25+ years here I have never seen one, only a common and harmless grass snake) however the NHS still records roughly 100 cases of bites in a year. They are more common to heathland and sea cliffs. Be wary if you have a dog out with you as there are reported dog deaths and dogs being bitten by Adders in the UK as well. They will likely only bite when feeling threatened, and if you do see one in the countryside, just be careful, avoid handling and let them be. (I have now actually seen one! Slithered across the grass when walking the dog, 26 years it took to see one!)
Deer should be treated respectfully in the UK if you happen to see any of the species (Personally saw a young Roe deer buck holding some rutting ground in my local forest.) We have Red, Roe, Fallow, Sika, Water deer as well as Reeve's muntjac. Some of the larger deer obviously have antlers over the autumnal months and can be aggressive during the rutting season but the likelihood is when out walking they will run from you - the issue here is being careful at night when driving they can come running out across the road (happens to me at least once a month) and they can go through a windscreen if hit. Also for dogs not used to deer, chasing them (also known as worrying) can lead to conviction and fines.
Badgers should be avoided whilst out night walking, again more commonly when out with dogs, due to their ability to destroy footpaths / walking routes with their dens, creating risk of a twisted ankle or worse. Near where I live badgers have almost completely collapsed a footpath by digging dens underneath it, which is something to watch out for. The likelihood of seeing one would be slim however, but dogs who like running down holes should have an eye kept on them. As a note they also are likely to be a bTB carrier (bovine tuberculosis).
Cows and Bulls are however a very common cause of injury when out in the British countryside. Cows can stampede when they feel threatened, especially when they have calves in the field. There were 215 injured and 6 deaths recorded in the UK between 2008-2011 due to being trampled by cows. However a good farmer will always signpost a Bull field or when cows have calves - allowing you to enter 'At your own risk' or leaving you to take another route.
Wild boar were reintroduced in 2004 to the Forest of Dean, there have been reports of damage to gardens, and attacks on walkers, dogs, and riders in the area.
Some people can worry about their dogs as Myxomatosis is a threat in the UK to British rabbits. It is not a threat to dogs or humans, but it can infect pet rabbits - so try not to handle wild injured rabbits if you see them unless aiming to take them to a vet or the RSPCA/SSPCA for treatment. And if you do need to handle them, anti-bacterial hand sanitizers will help prevent spread of infection.
Fleas and ticks are also common both domestically and in the countryside in the UK and they can be nasty for both humans and dogs, and Lyme disease can be passed onto humans by ticks.
There is also a debate / request you can find online and that comes up in wildlife searches for the reintroduction of Wolves to Scotland - however currently there are no wolves in the UK.
Edit: Seagulls can also be a pest when eating along the coast as they can swoop down grab food from your hand, as they can get quite large these birds shouldn't be hand fed or encouraged.
In addition to the wildlife hazards mentioned in the other answers, swans and geese can be intimidating even if not highly hazardous - the adage about a swan's wing being able to break your arm is said to be just about true, but unlikely - and walkers occasionally report hostility or aggression from landowners who dispute their right to use a particular route across their land.
To add a couple of non-wildlife hazards:
Avalanches can occur on British hills in sufficiently snowy conditions, so winter hillwalkers should be aware of how to identify and minimise this risk
Some countryside areas are used as Ministry of Defence firing ranges (marked as 'Danger Area' on Ordnance Survey maps). Where footpaths cross these areas there should be clear warning signs and systems such as locked gates or flags to prevent access during firing, but walkers should be aware of the possibility of unexploded ammunition. (Public access to military areas)
In general the British Isles are very safe. Most paths are well marked, dangerous areas are also marked off, and there are no large predators or other animals that are a serious threat to humans.
The other answer already covered this well.
Essentially there are no animals in the British isles that are deliberately dangerous to you, although there are a few that may injure or worse you if they feel threatened (and even those are rare). You're far more likely to be killed by the weather than anything else.
Situations to avoid:
Most paths are well signposted but on coastal cliff paths erosion is always a risk. Don't go too close to cliff edges unless you know it's safe as they can erode underneath and then collapse.
Weather can change rapidly and you should be prepared. Particularly if you are going mountain climbing make sure you take appropriate supplies, check the weather forecast and have a decent map and good walking boots. Even experienced walkers/climbers often have to turn back because the weather turns against them so do not feel that you can just press on - people die doing that.
General etiquette is to leave a gate in the situation you found it, if a gate is open leave it open. If it's closed then leave it closed.
Not sure how hazardous they are, but wasps, bees, hornets etc could spoil your trip in sufficient numbers or if you have an allergy to them.
The NHS has a good page on biting/stinging insects, how to avoid them and what to do if you don't:
A few others I could think of:
Foxes are generally not aggressive are easily scared away by humans. However, there have been cases of fox attacks of on small children. It should be noted these incidents are very rare. Interestingly this appears to be more of an urban problem possibly caused by the close proximity between humans and foxes and the apparent trend of people feeding foxes in their gardens.
Dog attacks are probably one of the larger 'wildlife' risks in the UK. According to this article there have been 5 deaths from dog bites since 2005, estimated >200,000 incidents per year resulting in approximately 6,000 hospital admissions. Recent legislation has increased the maximum sentence for dog attacks which cause injury from two to five years. Anecdotally I'll also add that dogs often try and run in front of my bike while cycling which could be another hazard.
Birds of Prey
There are many species of birds of prey in the UK some of which have been relatively recently re-introduced after becoming extinct due to hunting. Some are quite large (Sea eagles are the largest with a 2.5m wingspan. There are various reports of birds of prey attacking lambs. This has caused some unpopularity with farmers with reports of some birds being illegally poisoned on killed. There have also been fears that eagles may attack small children although there have been no reported cases and the RSPB calls these claims 'alarmist'.
In addition to the other excellent answers, you should be aware of the dangers of the infamous Scottish midge.
Before I destroy the Scottish tourist industry, in most places and seasons midges are absent or a minor irritation. But in the wrong place and time they can be horrendous.
Know your enemy, and go prepared with sufficient protective clothing and repellent.
I have twice seen people who needed medical treatment for toxic shock when they underestimated the seriousness of a midge attack.
Once on Skye I woke to find that my green tent had turned brown - the entire surface was covered in a seething mass of midges.
Other answers have covered the local fauna pretty well, but what about flora?
The only common venomous plant in the UK is the common stinging nettle, easily recognisable by its triangular, serrated leaves. This has fine hairs on its stem which, when brushed, pierce the skin and inject a mild venom. The resulting pain and swelling can be eased by rubbing it with dock leaves or dandelions, which often grow nearby, or applying an antihistamine cream.
Fortunately, allergic reactions to nettle stings are rare.
Very similar in appearance is the deadnettle, which is actually a variety of mint and does not sting. The easiest distinguishing feature is that the deadnettle has small white flowers on top, while the stinging nettle does not flower. Like all mints, the deadnettle has a square stem, and this is another distinguishing feature. However, the characteristic scent of some mint species is not shared by the deadnettle.
There are also many plants which have thorns, including thistles, brambles, hawthorn, and roses. These are not difficult to identify, and not especially dangerous beyond the obvious hazard of sharpness. With due care they can be bent back to make a safe path.