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.
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 agressive 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. 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, but it is not a threat to dogs or humans, but is to 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 for treatment - and if you do this, 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.