It's in the hospital's best interest to collect the money themselves, so if you explain your situation to them, and tell them you're willing to make payments even from the UK they'll be likely to take it up. It's more expensive for them to send it to a collection agency since the money that is collected has to be split between the hospital and the collection agency in the first place (or so I believe, that's the case between clinics/private doctors and collections). They'll want to keep the money for themselves, and if you push you can probably get a plan out them, especially if they have a patient assistance department of some sort.
Example: I had a lump that I needed checked out (yeah, I was dumb and freaked out and thought it was breast cancer since my mom had it, but I was scared) but I didn't have insurance. I called an emergency room and asked if they took uninsured people and they said yes, so I went in. I was checked in the emergency room, and even given a referral for a surgeon to follow up, and it was emphasized that I would be able to work with their financial department to work out a plan since they knew it would be expensive.
Healthcare is expensive for many factors--there's so many things that goes into emergency/urgent care visits and it does add up quickly, and it can be overwhelming. But the buck only gets passed on to everyone else if a bill goes unpaid, and a hospital wants to avoid having to send everything to collections. Unpaid bills do contribute to the elevated cost of healthcare over time, once hospitals and doctors try to fill those holes in their books. They defintely can't take it in the shorts by not getting paid at all, and it's much cheaper and easier for them to collect it directly from you, even if it is payments, as long as they think they *will* get money.
Ninja edit: you may have to talk to more than one person; some can have a stick up their bum, and some can just not be aware or familiar with exemptions that can be made to whatever magical number rule they have. I'd like to think they'd be flexible in order to get their money once they're made aware that you're from overseas. The first lady might have given you a collection's speech, but I'm pretty sure someone there will compromise with you. Like I said, it's in their best interest to compromise, not to be stubborn.