Yes only one personal statement which is sent to all your choices. Your application is read by professors in the department who will be teaching you. They are looking for students who are committed and passionate about the subject they will be studying. Your PS needs to show this commitment and passion towards Econ/Finance/Law in an academic sense, with relevant ECs referenced. Lots of info out there on how to write a good PS.
I think the lack of AP Calc will be problematic at any of the UK Econ/finance programs. Have you looked to see what UCL and Warwick require from A Levels? AP Micro/Macro are seen as not that great. I think Oxford uni website has a list somewhere of equivalency APs for A Levels…
UK universities have very little / no handholding of students. You are expected to do a lot yourself as an adult. Are you sure that is what you want and is it what will best help you thrive?
There are excellent schools that would prepare you for IB that might offer merit in the US: Kelley - Indiana, SMU - Cox, for example. My daughter is at a big southern flagship on merit and knows people going to Goldman Sachs, front office. Plus there is a large regional IB market, such as in Houston, which SMU Cox would feed into.
And then there is the whole Q of do you need to do an MBA after a couple of years etc etc, which costs more $$. You could go to one of these cheaper places, do well, work hard, get your top tier MBA and enter bulge bracket IB.