Since you're in the UK, some UK-specific advice might be useful: there is no law in the UK to state that only men can use the Gents' and only women can use the Ladies'. So you're already entitled to go wherever you wish. You can use being baby-faced and your resemblance to a 12-year-old to your advantage: remember that boys use the men's room too, including baby-faced 12-year-old boys. So if that's what you look and sound like, you already look and sound like loads of people who regularly use the men's toilets.

A good way to do it is to build your courage by slowly working your way up. First, decide to stop using the ladies' altogether. Then, start using gender-neutral loos (e.g. most branches of Costa) or accessible loos (they're
not disabled loos, they're accessible loos for anyone who needs more space, easier access or more privacy!) wherever possible. Then start using the least busy men's loos before working your way up to the busier ones - even if it's just to pop in to wash your hands at first. Many branches of Starbucks have single-stall men's loos, which is a good way to introduce yourself to them. So do some branches of McDonald's. You can then work your way up to multi-stall affairs in major shopping centres. As for sanitary products, just wrap it in some toilet paper & put it in your pocket; you can then throw it away in any public rubbish bin after you leave. If you're worried about the noise, there are plenty of guys who need to use things like Tena, and they do this in the Gents'. You could even flush the toilet to disguise the noise.

I started using the Gents' about 8 months before I started T, and around 14 months before I started to pass at all. And yes, my GIC was very happy with the fact that I was doing so and I'm sure it's one of the things that sped up my access to T and surgery. I was very nervous the whole time, but I was never challenged once. I just marched into the Gents' like I belonged there; kept my head down; did my business; washed my hands and got the heck out. That's the beauty of the Gents' vs. the Ladies': guys don't look at each other like women do. The reason is due to internalised homophobia, which is deplorable... but at least it means that they're less likely to pay attention to you.
As for gym changing rooms? You still have the right to use whatever ones you want, but if you're pre-op you may feel self-conscious about working out, showering and changing. Personally, I didn't use the Gent's changing room until after I'd had top surgery. Instead, I spoke to the manager at each of my gyms, explained that I'm trans and that I didn't feel comfortable using the ladies' changing rooms but didn't feel safe using the men's yet because of my chest, and I asked them what they could do for me. One gym arranged for me to use the accessible changing room, the other let me use their staff changing room. They were super-professional and discreet, and I felt much more confident in using the gym because I knew the manager would support me if anyone ever kicked up a stink. Then after top surgery I went back to the managers as a courtesy to let them know I was going to start using the men's changing rooms, just in case I ran into any problems. I needn't have worried though; I've had no problems whatsoever.
When it comes to exercises, I was pretty dysphoric about my pre-T, pre-top-surgery body so I'd pick exercises that wouldn't make any unwanted parts jiggle (e.g. cycling is a good alternative to treadmills) and I wouldn't work out in front of the mirrors. I don't recommend binding whilst working out, so I'd just use a sports bra and I'd wear a hoody over my T-shirt & hope for the best.