I have a Sonny and Brownie CD entitled 'Back Home Blues' and the second track is 'Whoopin' the Blues' (Sonny on his own) which sounds like the one on TV advert. I remember hearing this paricular piece of music when I was a child/early teenager(?) on Two Way Family Favourites on a Sunday Lunch time. I'm sure Burl is right about Jack Jackson playing it on his programmes.
In the late 50s, Sonny and Brownie spent a lot of time in Britain, touring with the Chris Barber band. As 'singers/variety artists' they weren't subect to the Musicians' Union ban on Americans playing here, and I was lucky enought to see them three or four times. Whoopin the Blues was one of the standard numbers in their set.