Regarding your debts following you to the UK...as far as I know, debts are limited to a single country and cannot cross international borders. So no, they could not garnish your BF's wages for your debts, even if you're married, assuming you are residing in the UK. (The caveats here are that: 1) I'm not an expert 2) I don't know what sort of protection you get - or don't get - as a non-UK citizen, though I'm fairly certain they can't touch your BF if he's a UK citizen.)
However, if you ever decided to move back to the US, your credit would be completely shot if you had many debts go into collections/go ignored for months/years. I would strongly recommend having a definite plan for how to pay back your debts so as not to limit your options in the future. (Best case scenario: you decide you want to move back to the US together for various reasons; worst case: something happens to the relationship and you move back alone, something happens to your BF and you move back alone, your living situation there deteriorates to the point where you have no choice but to move back, etc.)
I would recommend calling your creditors and discussing your situation. Provided they have no way to garnish your wages, you should be able to work out a deal with them...even if you just promise to send them $5/mo until the debt is paid off, if you send that $5/mo in to them regularly and on-time, they can't really complain (unless they find out you're making $100,000/yr or you start buying yachts or something).
You can start work right away on a spouse visa. As far as how difficult it is to find a job - it depends. The economy in the UK is (arguably) just as shot as it is here...if you're just looking for a cashier type position, it may be easy to find (esp. in large areas like London with lots of retail) or nearly impossible (in small towns with a handful of shops).