My first 2 bunnies (Izzy and Jake) were adopted via pets at home. The one I used to go to (I don’t live in the UK anymore) had a charity set up inside.
As far as I know, it’s an actual charity, so would be the same as adopting from a shelter. My two came with reasons they were brought to the charity in the first place and a personality chart (Izzy was a Easter present for a child, family got bored and didn’t realise how much work rabbits were. Jake was adopted 3 times, going backwards and forwards because he had a history of biting. Before we got him he was given up due to developing snuffles and they didn’t want to pay for vet bills) there was a “suggested adoption fee” and the charity also helped to pay for them both to be fixed (a voucher for steri and a written agreement to get them sterilised at the time of filling in the paperwork)
The charity also found loving homes for dogs and cats too, but was only a few smaller animals that were “shown”