Often in British countryside you'll find fields with horses in (and cows and much of this is applicable to bovine) where it is unavoidable or rather difficult to go a different way and the are a few things to be aware of. Usually you'll be heading over something similar to this:
As public rights of way here allow footpaths through any field, garden, park, etc etc. I've walked through many a front or back garden whilst people are sitting out in summer (Kinda awkward!) but it's how footpaths work here.
Firstly if you do come across a field with horses in, before entering the field take a good look around you, if the horses are far away, then enter the field and keep yourself, any children, or dogs close to you, with dogs on a lead (though child leads are not uncommon for toddlers!). Walk around the edge of the field to the other exit. Try and keep children and animals quiet too.
Pretty simple if that's the case, but not always will it be so.
If the horses are scattered, stick to your chosen path around the field. Don't think to cut across unless you feel confident and it's not a large field.
If it's muddy, which is common, try not to make a lot of fuss, I've seen horses spook at their own shadows.
If the horses are close to the gates, never fear. Actually that's a valid point and not a saying. Don't act skittish, keep your head up, don't make eye contact and sensibly navigate the horses keeping to the front, a bite is better than a kick.
Always avoid approaching the horses where possible, and definitely don't bother feeding them (most likely they're going to be on strict diets and an apple from every stranger won't help!).
If you're cycling through horse fields the above is all completely applicable, but the best advice I can add is to dismount and walk around calmly.
Also bear in mind that in the UK people can possibly be riding in these fields, I used to gallop down the fields in summer and often had to deal with people who did not look around before rushing out into our fields.