Getting a tattoo as a souvenir of a trip, is to me, not a very smart idea, but many younger kids these days (millennial's) have a completely different viewpoint regarding them (tattoos). I suppose small tats would make a better memento to remember things like a major vacation.
Tattoo parlors everywhere have bad reputations for cleanliness and distribution of diseases, but if you are looking to get one, I'm sure you already know this. In England, especially London, there are many up-scale parlors who do walk-ins. Just do some google searches for tattoo shops and the city of your visit. You'll want to find artists that match the style of what you are thinking of. Parlors in London are (essentially) the same as in the US. There are a few high-end shops, and many low-end ones. Do yourself and your health a favor and visit the high-end ones. They'll have the best reputations for health and safety. There isn't anything "going around" in England right now, that isn't a concern everywhere else (such as Hep).
If, by France, you really mean Paris and its environs, that is a different deal. French shops are a bit more controlled by the state, and NONE of the higher end shops there take walk in visitors. They all require appointments and many lists are a year or more long. Do some on-line research and try to call them ahead of time, if you have time. They're not going to want to spend time with language barriers, though, so be prepared, or have a (French-speaking) friend with you. In fact, I wouldn't go to a shop in Paris that didn't have a waiting list, simply because those are the ones you want to avoid.
Good luck, and happy travels. You'll end up with souvenirs to talk to your grand-kids about.