My father's accent was mostly public school, my mother's was fairly nondescript southern England somewhere - certainly nothing strong like her sister's, which is more rural Hampshire.
I sound like I stepped out of a Hardy novel. On a daily basis, I tend to modify it (only partially consciously) towards RP, as most of the people I speak to are not native English speakers, and if I were speaking thick Dorset with a load of dialect words, they would have no idea what I'm on about.
When I was first out of school at 18, I had to have someone interpreting between me and a Brummie guy I met (though to be fair, there was also alcohol involved.) I've also had an American interpreting between me and someone based in Hong Kong on a 3am work call (I could understand her, but she couldn't follow me, but as it was 3am for me, I can't be sure I was entirely coherent anyway.) So I know my accent is sometimes a problem, and it's something I've found confusing at times, because I am quite good with other accents (which I think is partly because I was brought up on Radio 4 and listening to lots of different voices.)
I do occasionally slip when I'm wound up about something, and my accent apparently gets stronger when I go back home or when I am talking about Dorset. I can also slip into a Welsh accent quite easily, as I spent a lot of time there as a child - and am currently learning Welsh. I remember my grandfather teaching us how to pronounce the "ll" sound, too. I've learnt other languages, and although I can hear the accents and I don't mine is too bad, though never fluent, I suspect I might speak them all with an underlying hint of Dorset.
Anyway, I do think you're unlikely to have that much influence on your children's accent once they're at school.