Have had 5 cats from Wood Green in Heydon.
They'll ask about your home environment etc to ensure the cat(s) in question fit in .... eg. do you have any other animals, ages of children, garden, busy road etc.
They'll then go through the animal's history with you, including any odd habits it has or medical conditions.
I think from memory, they ask for a "donation" (that's what they call it but they are quite assertive about it - not that I think that's wrong) of £50 for each animal. The cat will either already have been spayed/neutered, or, if it's too young, they give you a voucher (I think) for you to take to the vet when it's time to do it. Every cat will also have been micro-chipped.
All this was done on the same day.
I've never been to the Godmanchester shelter, but at Heydon they have white boards in each run which briefly describe every cat - age, sex, colour, "doesn't like kids/dogs", "must be homed with so and so other cat", so you get a brief idea before you go into the office to discuss the adoption.