BarbaraB I agree. I coudn't find any info on spunging houses beyond a very small snippet or 2 on the Net.. (havent time to look for books). But my feeling is that its unlikely that they were "only for a certain class". It would be in the interests of everyone to get the debt paid quickly, so I'd imagine that anyone vaguely likely to be able to get the debt paid would say "Please, hold off a minute. Let me send a message to my friend or my former boss, or my father or (in Eliza's case) her "dear Friend Mr X" and he'll come and pay the debt."
And I would imagine that the authorities woudl prefer to do that and give them a couple of days grace to pay than to get into more complicated situations involving courts and debtors prisons.
Brandon's servant was possibly someone who had left his service and tried to set up a business or gone inot another job, and lost the job and got into some financial trouble. THen he would naturally remember his kind employer who would certainly try to help, and Im sure that they would give him that 24 hours or (more realistically I think 3 days) to send to Col B and wait for his reply. Eliza was a "fallen woman" true, but she would have problaby had SOME friends who had money, (possibly well to do "bits of muslin") or former lovers, who might help her out.. and since she also would have sounded like a lady, she would again problably have been able to say with some degree of conviction that possilby X or Y would help her out and be granted a few days' grace. And certainly It woudl be possible for her to have her little girl with her..