Normal in the UK is applying with “predicted” scores for exams taken in June, with results coming out in August (yep- August, before term starts in Sept/Oct).
The person who writes your LoR “predicts” the scores for your APs, and the uni gives one of 3 replies to your application: rejection, “conditional” offer or “unconditional” offer.
If you are applying with “predicted” scores (and you aren’t rejected outright) you are likely to get a “conditional” offer, which will say something like "we are pleased to offer you a place on X course commencing in Sept/Oct, conditional on the attainment of scores 123, on exams XYZ.
A lot of Americans accept something in the US, to have an option in case they do not make their offer (ie, the scores don’t meet the condition(s)). and then just forfeit the deposit if they make their offer.
As for whether you would get an offer from either of these programs in the first place…I’m assuming that when you say AP Calc you mean BC, as Bath requires that, and that you have a portfolio that meets Sheffield’s requirements. Your PS should be highly academically focused.
In general, UK unis like US $$ and an applicant who meets the requirements has a high likelihood of success. However, some courses are less predictable, and Architecture is one of them- for a start, it is highly competitive, but equally, they want to be sure that you are able for the course, and art/design background/abilities are tougher to assess.