I don't have experience with UK and US gymnastics, but I have gathered a lot from others over the years.
Prices:
like the previous commenter said, the prices vary across the US a lot. From what I hear, it's WAY more expensive than in most west european countries.
Training type:
The US has two most used systems. DP (also known as JO), which uses levels 1-10, and excel which uses metals/minerals.
---DP is generally the most rigorous with more hours. Compared to many other countries, and compared to GB zinc, DP is a lot more rigid in their skill expectations. Where GB zinc lists things like "Acro elements forwards / sideways &backwards", level 4 specifically lists skills that need to be done, like "front handspring stepout" and "back walkover". In the higher levels (called optionals), this changes and the DP code becomes more free/open.
---Excel is a lot more free in its code from the start, with requirements like e.g. on beam "2 acro elements, of which 1 through upside down vertical". This might be more like what you're used to. However, Excel was also designed to be the lower intensity option originally, for kids that want to spend more time on other hobbies etc. Not all gyms take Excel equally seriously. Sometimes they don't focus as much on good form, it'll really depend on the gym.
Level:
I've had a quick look at the requirements of zinc and the rough requirements of the US levels. I think zinc is a bit higher than level 4 or excel gold. Specifically, zinc requires a higher cast on bars, bigger leg spread in split leaps, and a backwards element on beam. Those aren't required in DP level 4 or excel gold. However, in zinc you don't have to hit all the requirements, so that evens it out a bit? And zinc vault is a bit easier than DP level 4 / excel gold I think.
In the end, it really depends what skills your daughter has, and what the gyms you're moving to are willing to work on / what they require. If she fulfills her back acro requirement on floor with a backwards roll and can't do any other back tumbling, she'd have a harder time going to a DP level perhaps.
If you want to know what level she is, list her skills for yourself honestly, and see what US level fits that best. You can get a general idea of the levels requirements at:
https://gymnasticshq.com/gymnastics-levels/ but do keep in mind that its not perfectly accurate perse. And sometimes gyms also have their own (higher) standards/requirements.
Terms
I notice some terms are different. Just so you know
upstart = kip
spin = turn
Straight = layout
Salto = usually just indicated by the body shape (eg. backwards salto tucked = back tuck)