Trust me, you will have an accent even if it's very mild. Someone with keen ears somewhere else on the country will be able to tell. Unless you live in Downton Abbey, it won't be RP. RP is proper King/Jacob Reece-Mogg/1930's BBC continuity announcer stuff. Been trying to hide mine for years as was always told by parents and teachers that the local accent makes you sound "thick" so try to tame it as much as possible. Now I pretty much sound like that Professor Alex Cooper (but obviously not saying clever stuff).
I definitely don’t have a “proper”, King’s English style RP accent, but I’d say I’m definitely closer to RP than any other regional accent in the UK. My whole family and most people who’ve ever met me say I’m “accentless”, I don’t have a rhotic accent and don’t roll my r’s at all, unlike many people in my area, and I use the more South East-style long a (bar-th rather than bath) so I’m not really sure what else would give me a South West accent other than maybe some little, minute “isms”. I don’t think I have a
strong or particularly noticeable South West accent, put it that way.
I once watched a video about British regional accents, and there was one described as “contemporary RP”, which is basically RP but slightly less formal than “proper” RP; a generic British accent, really. I thought this described my accent quite well.
None of my immediate family have overly strong accents. My sister, like me, is pretty accentless. My dad is quite accentless but has the odd London/South East speech quirk from having grown up in Kent; for instance, he occasionally pronounces the “th” sound as “f” and doesn’t pronounce the “h” sound, so as an example, “your health” becomes “your ‘elf” and “truth” becomes “troof”. He occasionally pronounces my full name, Matthew, as “Maffew”. My mum probably has the strongest South West accent out of us, but even hers isn’t ridiculously strong; she describes her accent as “an eclectic mix of Newport, Bristol and Forest”, and I wouldn’t say my mum had a properly strong Forest of Dean/South West accent compared to some people around here.