Why do I not appear to have a regional accent?

Without knowing anything whatsoever of your circumstances, OP, I’ll guess you’ve picked up some supraregional dialect koine somehow.

Like, I dunno, RP, or whatever has replaced RP in England these days.

It’ll have a lot to do with your upbringing and your socialisation, as others have said. This kind of accent mixup is very commonplace in children of military personnel, who move around a country frequently; hence the term “army brat”. And of course prestige variants of a language are produced all over a language community, unified by ideology or class rather than regional identification—even if their origin is often regional.