Larry -
If the OP had taken her cat for treatment on Friday night, the vet would have discussed her options. Kidney failure is one of the first things checked for, especially with the symptoms described. The OP would have been given the options, including prices, and an explanation that by Monday the cat would not be saveable. If it was the OPs decision that they'd rather put the cat to sleep, the vet would have done so. Any treatments would be fully discussed, with prices, so the OP could decide what she could pay for.
Repayments are typically very low. Usually between £8 - 15 a month. Most vets can't offer credit agreements legally and therefore can't affect your credit rating anyway, which is why they aren't keen to accept a repayment plan. But they would have done, because the alternative was to treat the cat for free. The vet could not have left the OPs cat because it was cruelty.
The OP didn't say she'd taken her cat and needed money for treatment. People do donate frequently and generously in those circumstances.
There are also places you can seek help, although they tend to not be well advertised because of the huge amount of people who contact them. I'm not talking about the PDSA, but charities and organisations and vet chains who will treat pets for free or a very low fee. The vet would have looked into this once the OPs inability to pay became obvious.
The veterinary world is far from perfect, I know that. But kidney failure is painful, and absolute agony to watch. Even cats who are being treated for kidney failure will make you sob. They are in pain. They feel like they are dying, and they will cry, and fit, and attack themselves.
If someone bought their cat in that state to any vets, I'd bet my life that the vet would treat it regardless of the financial situation. That can be sorted later.
And cats can be insured from £3, and are generally low-cost unless they are an at-risk breed, so that's extra peace of mind for the owner.