Adoption / rescue cat question

I currently have a 2 year old male rescue cat that I adopted from a shelter when he was only 10 weeks old, he had been abandoned and left to fend for himself. I’ve lived in my current home for about 3 years, over the last year I’ve saw a black cat roaming around at all hours of the day and night and originally assumed it was a neighbours cat, a few months ago I saw her wandering at 3am and popped some food out just incase and she scoffed it down so I done this for a couple of weeks and eventually I managed to put a paper collar on her with my number and a message about if she’s yours please contact me so I know she’s not lost and never heard from anyone though she had it on for a couple of weeks, I continued feeding her throughout the weeks. She wasn’t very comfortable being too close to me but she showed up every night at the exact time to have something to eat and waited for me to put her food out and eat. I made an outdoor bed for her and a shelter for her food. Ive put her photos on at least six different missing pets pages and people reply often saying they’ve seen her in my area but don’t know who she belongs to. I put a second collar on her and still never heard anything, the last few days she’s become a lot more friendly with me she approaches me now and rubs against me, today she sat down on my knee and ate from my hand and was purring. She’s attempted to jump in my window a couple of times but never stays more than 5 minutes. I’m wanting to take her in and care for her, I hate to think of her outdoors alone and cold. Is there a way I need to go about this? Do I have to call a cats protection company first? Do I have to have someone come out? I would take her to the vet to be scanned and checked over of course and if she does belong to someone I’d attempt to help reunite them but is there a specific process I have to go through? I can’t just take her in my home and keep her? Any advice would be great!