The aunt may well have lived elsewhere in Devon, hence they "left London to stay with an aunt in (somewhere-not-Teignmouth,) Devon, and within Devon they eventually moved to Teignmouth"
I don't mean to be ungenerous, just pointing out a very small detail the author may not have been aware of.
I agree with your reading of the Wikipedia article, which I take to mean that they moved first to somewhere else in Devon, and then ended up in Teignmouth.
The re-wording is slightly less precise and does not offer more information.
It is re-written in the article as:
"In his earliest life, he and his mother moved in with an aunt in Devon, then to Teignmouth, and then finally to Bracknell on account of wartime bombings."
As I said, it really is only a small detail - and completely forgivable as I'm sure many people have no idea where Teignmouth, or even Devon is. However, on current reading it would be akin to saying "moved from England to London"
The aunt may well have lived elsewhere in Devon, hence they "left London to stay with an aunt in (somewhere-not-Teignmouth,) Devon, and within Devon they eventually moved to Teignmouth"