Would I be able to recruit into London from the William and Mary - St. Andrews Joint Degree Program?

Firstly, there's no real OCR in the UK, only informational visits. Everyone applies online via the same system. Summer internship applications happen September/October of penultimate year, spring weeks November/December of first year - but in your case, you'd apply second year as it's a four year course. Try to get a spring week if at all possible. If you're in the UK (with work privileges that a study visa grants), I don't see why the W&M degree would make a difference. If you lose the right to work in the UK while at W&M (which I imagine you would) then you won't be able to easily intern that year in London. Worth speaking to the faculty about re. visas as that's the main issue.

Secondly, you seriously overestimate St Andrews' reputation within finance. Academically, it's a great school and attracts some very talented students - especially Scottish home students, who will often choose it over Oxford, Cambridge, Imperial, and LSE because of fees. Its layman's reputation is reasonable; most would put it in the UCL/Warwick/Durham/Edinburgh category. However, it's relatively small, and very remote. It therefore has minimal placement into the City. Even Edinburgh has minimal placement, just by virtue of being Scottish, but it's an even harder task coming from St Andrews. Most banks will take ~90% of their class from LSE, Oxford, Cambridge, Imperial, UCL, and Warwick. The rest will generally be top of their class at Bristol/Bath/Exeter/Nottingham/Durham/KCL (and sometimes Edinburgh and Manchester, but there's a distinctly southern bias except Durham), usually (but not always) studying Economics or STEM. St Andrews won't be impossible, but it's an uphill struggle - especially if you're not majoring in something highly quantitative. I'd think about whether the joint degree is worth it if you want to go into finance.