Many years ago I lived in Battersea
I think I would check for train noises more than once just because so close to the junction itself not just a line leading out of it. BUT that said IME lower ground rooms are very well protected from noise either traffic or trains so given that is where the bedrooms are - I would not worry about having a good night's sleep but I would visit more than once to check general train noise - it may well be fine.
It looks very convenient - I would say it is not the "best " part of Northcote road area ( the roads off Northcote road itself tend to be that & if you want to go out & don't want to drive how close are for walking either to Northcote Rd or elsewhere ( although that general area I think is well served for buses or indeed a short cab ride.) The Winstanley estate used to have a bad reputation , but I do know the council earmarked a lot of money to getting it refurbished / upgraded - so my knowledge is probably out of date.
I am sure you have but I would definitely go and have a good walk round in all directions and see how you feel about the area. Other than that I think the flat is great.
The flat itself I think is lovely & for the area a fabulous garden . I think being the freeholder would give me some comfort ( although there will be responsibilities along with that. ) I am not convinced all modern flats are better soundproofed - some are some aren't .
Good luck @lilyannofpembleton 