There is no published literature on this particular case of lexical variation.
Our investigation has uncovered clear patterns for this particular case of lexical variation. Roll is the most popular choice for speakers in the UK and is unsurprisingly widespread throughout the country, although it seems to be slightly more prominent in London and its surrounding areas. Tea cake is almost exclusive to the North West, particularly areas north of Manchester, like Leeds and Blackburn. Barm is another variant used almost exclusively in the North West, in a widespread region covering the likes of Manchester, Wigan, Preston, Liverpool and Blackpool. Muffin is another Northern lexical choice, particularly the areas just north of Manchester such as Oldham and Rochdale. Cob is only used by a small fraction of the population, but these speakers are largely concentrated in the Midlands around Nottinghamshire. Bun is another term that is used by speakers all over the UK, although interestingly it is favoured by speakers from the North East; this is the only region where bun is the most popular choice. Only 81 respondents use the term batch, and these speakers are mainly concentrated in both Liverpool and Coventry.