Melysion, good for you, and thank you for wanting to help these cats!
They need every person they can get on their side.
There's a sticky at the top of this page called How to Trap Ferals and More, which will give you much better information than I can. I'm new to TNR myself, and not any kind of expert. But here are a few things you'll definitely need:
1) A humane trap. The brand everyone seems to use in the US is called Havahart. I'm not sure what your shopping options are in the UK, but if you have something like a Home Depot or Lowe's, check the pest control department. (The same traps are also used for raccoons and possums.) You should also be able to find them online.
2) A place to keep the cat. They are extremely hard to handle, so you may want to just leave them in the trap for the whole process, if you can get them into the vet quickly and plan to release them right away. If you need to keep them for awhile, you'll need a carrier or crate that's big enough to keep cat + food + water + litter box in. You'll also need a room, garage, shed, or other place to keep them (while they're in the carrier) that's separate from your housecat(s).
3) Typical cat supplies: old towels or blankets to lay on and use for cover, food/water dishes, litter box, etc. And some litter suitable for post-surgery. My vet recommends a brand called Yesterday's News, which is little pellets made of recycled newspaper.
4) A place to get the cats fixed. If you're also going to be working with a shelter that will do it for free, great! If not, you'll have to find someone who will fix them for a non-bankrupting cost.
As for how to tell if a cat is truly feral, sometimes it takes a few days of observation and interaction. There's a whole continuum of behaviors between "truly feral" and "tame housecat." Most of the ones I've dealt with seem to fall into that gray area, so use your best judgement. But if you aren't rehoming them, and you're getting the surgery for free, I say TNR whomever you can. If they belong to other people, tough! You've still helped society, and they're getting a free service. Even if they're mad, chances are they'll never know it was you.
There are a number of TNR veterans here, and I abuse them regularly by demanding reassurance through every step of the process. I can vouch that everyone is really patient, friendly, and helpful! And if it will help, I can contribute moral support from one newbie to another.
Good luck, and again, good for you!!!