I find a good way of avoiding the issue of someone having bought you a present when you don't have one for them is to get (and pre-wrap) a few small items (like a small box of chocolates, say) and then you have them handy if you need them, and can just produce something and say 'thank you, and here's your present' if you unexpectedly get given something. (And if you don't end up beineg them you have loads of chocolates left over, so result!)
As a rough guide, I'd say grandparents/parents/children, siblings, partners and close friends would usually exchange gifts, and then depending on the habits and preferences of those involved this might extend to more people. For example, I usually get two or three presents each for my parents and my brother and his girlfriend, as I spend Christmas day with them; and one small-ish present each for the aunts, uncles, cousins and their husbands and children on one side of the family (who we used to spend Christmas with), but not for those relatives on the other side the family (because we've just never done that). I also usually get things for my close friends, especially the people who feed my cat over Christmas; for my downstairs neighbour (to be polite), and the people I work with. But, a lot of those will be small token gifts (for example, at the moment I'm making lots of jars of fruit jelly with wild hedgerow fruit so I can give them as gifts at Christmas.)
Secret Santa is quite common, especially in workplaces or between flatmates - a group doing Secret Santa would often exchange their gifts at a Christmas party or meal in the run-up to Christmas.
If you will be in the UK on Christmas day and make any plans to spend the day with other people, you may want to think about whether to exchange presents with those people. If, say, you were invited to someone's home on Christmas day it would be appropriate take them a gift (and I would definitely get a present for any children I was seeing that day). If you're hanging out with friends that day it's fine to ask if you're 'doing' presents and talk about what you all plan for the day. Oh, and be aware that EVERYTHING is shut on Christmas day (except the odd convenience store, or some pubs/restaurants where you can get Christmas dinner if you pre-book) so it's a good idea to a) make sure you have food the house for a few days, and b) make some sort of plans, because actually would be quite a boring day otherwise.
Giving Christmas cards is pretty ubiquitous - friends, relations, colleagues and neighbours will all tend to exchange cards - but they're cheap to buy (most people just buy a few boxes of cards) so this isn't a huge commitment.