Hoho..! This is my floor, I think.
Been there, done that, successfully resettled many ferals in my time.
First.... don't even think of trying to grab a cat that has never been picked up.
It will think it's about to die and they fight valiantly to save their lives. Everything about feral cats is about the control of fear... cats naturally fear anything they've never experienced. That's why we always use purpose built and safe equipment for handling and transporting.
Use a proper, purpose built cat trap (not to be confused with the cheap and often lethal rabbit traps people try to flog you on Ebay
) which you can borrow from the nearest cat rescue group. They will tell you how to use it or even trap the cats for you.
When you catch puddy, place a big towel over him/her and take the whole thing to the vets. They will do the rest.
Second... don't just turn a feral cat loose in a strange shed. It will do the wall of death and scoot past you and out the door.
Invest in or borrow a dog crate (available cheap enough from Argos). Set up the crate in the shed and decant the cat into the crate. Leave it at least two weeks before one day just leaving the door open, then puss can go explore the shed. Another 2 weeks and leave the shed door open.
This gives puss plenty of time to get used to you as carers and to figure out where the food comes from.
I agree that you need a minimum of two cats. Cats in the wild live in family groups or colonies. A single cat is likely to go off in search of a pal.
Cats above all do not like change. It takes time for them to feel confident in new surroundings. Smells, sounds, food source... all these changes must be introduced slowly. Over do it and the cat runs away.
Of all the ferals I've resettled only one has not stayed around, so any problems feel free to PM.