4 months is a loooooong time in baby land - think of all the things your child can do know that they couldn’t do when 4 months old.
Don’t panic about “bad habits”. I worried so much with DD1 about sleep habits, making a rod for my back etc.... my advice is, if you’re happy and what you’re doing works for you, stick at it. If you become unhappy, change it! There’s absolutely no reason in the world for you to change something you are happy with.
I’d read a tonne of stuff about self-settling and was very worried that I was in some way depriving my child by not teaching her these things. I fed her to sleep because it worked well for me when I was on maternity leave - I could get her to sleep ANYWHERE in any conditions. I was worried about returning to work. But there was no need to be! Turns out, when I wasn’t there to be an option, DD didn’t need it. When my DMum had her, she’d take her out in the pram. First day or two took a little while, but by the third time (I think) it was completely smooth sailing (I know you’ve said you don’t want to do this because of cold - that’s fine, my DMum could also cuddle my DD to sleep, but preferred the walk and then having her hands free). With my DH, she napped lying on his chest (he liked doing it this way so he could relax a bit too). When she started nursery, her key worker held her on the first day, and on the second she just lay down on a mat like all the other children and went straight to sleep. No problems whatsoever. Children of a year old can learn that different carers mean different routines, and as long as they are happy and content with that carer there’s no reason to think there’ll be any issues.
Incidentally, I didn’t stop the bedtime feed till DD was 2.5. Much later than I’d originally thought, but it just worked like magic for such a quick and smooth bedtime, regardless of what level of light/noise there was. Had no trouble at all changing that when I wanted to, no tears or tantrums. And as a 3 year old, I kiss her goodnight and that’s it. She’s an amazing sleeper. So please don’t worry about bad habits - feeding to sleep works so well because it’s a very natural thing to do, and babies will grow out of it! And they cope just fine when it’s not available. Your mum will find her own way in a day or two.