Lots of really helpful advice here, Abi. The only thing I really want to add is that if Lucy takes a while to settle in, or seems nervous, try Feliway. Stress symptoms for a cat include some things that seem quite odd to humans, like going to the toilet in a place where she knows she shouldn't, and standing on her hind feet scratching up the walls. Feliway contains cat pheromones and will help her feel safe and at home. It makes a life-changing difference for both our cats: the nervous one stopped being nervous and even the confident one is more relaxed and happier.
The other thing that is worth mentioning is that if she does go to the toilet somewhere you don't want her to, it's really important to take the smell away. When you are new to each other she might not know where the right place is, or she might use the wrong place out of nervousness or stress. But once somewhere smells like a toilet, she will keep going there - at that point it's normal cat behaviour, not an indication of something wrong. There are lots of commercial products that claim to remove the smell, but in my experience only one that really does (at least as far as the cat is concerned, and of course her nose is much more sensitive than yours!). Luckily it isn't prohibitively expensive, but you will only be able to get it from the vet's or from the interweb, not from a pet shop.
This is the stuff. Make sure you get the one with a purple label, that gets rid of smells and stains.
I don't have shares in either of these companies. I'm just owned by two cats who have occasionally had problems.
In all that time, they have never increased the premium, apart from the usual small cost of living rise. The only thing you have to watch is that they do ask you to pay 20% of the whole cost of treatment when your cat reaches the age of 10, rather than the usual 10%. That caught us unawares after Abby's first op.