HMRC can't release any information about you to a company or anyone else beyond your tax code.
On the wider point of getting found out, sure if he was looking to get a job as CEO of Glaxo Smith Kline then they'd check everything they could (still can't access tax history though). For the job knocking together picture frames that
@richyrich was interviewed for it wouldn't go beyond a basic CRB check if that.
BS on a CV or interview is just a way to cut out other BS. You can either do a job or you can't, and I would suggest that anyone with a full set of limbs and basic hand skills should be able to staple gun and glue picture frames.