I don't know if NHS follows WPATH or similar guidelines, but those guidelines don't state that T before chest surgery is required. It can definitely be done first, especially if your chest is keeping you from passing as male effectively. I'd bring up to them that this is the case in the US if they try to tell you no.
As far as T goes, unfortunately you don't get to pick and choose what effects you get and what you don't. But the bad stuff doesn't always happen to everybody. I haven't experienced any hair loss whatsoever, not even thinning. Facial hair seems to be a slower change for everyone, so I wouldn't worry about that right away. It's not something that would sneak up on you. Once it started growing you could find a way to deal with it. Mine didn't start coming in well until just before a year.
T is tough. I know there are plenty of irreversible changes, and a lot of them can happen early on. But it's also the only stage of medical transition that you can easily stop if it's not for you. I'm the kind of person who will always wonder "what if?" if I don't try something, so I knew that I would at least try T at some point. I'm very, very happy with my changes BUT I also present very masculinely and actively wanted most of the changes that T brings.
While it isn't a surefire solution, I'd recommend starting on a low dose if/when you do. It isn't guaranteed to make things happen more slowly, but you'd have a better shot at it than if you started on a regular dose.
