There is no published literature on this particular case of lexical variation.
Our data shows clear regional patterns for this particular case of lexical variation. Barm is confined to the North West, comprising an area that runs from Manchester westward to Liverpool and northward into the western half of Lancashire (from Blackpool to Preston). Tea cake spans the eastern half of Lancashire (Blackburn, Burnley) and the western half of West Yorkshire (Bradford and areas around Leeds). Bread cake is also used in parts of Yorkshire, clustering primarily around the urban centres of Sheffield and Hull. Muffin is perhaps the most geographically localized, confined to East Manchester and areas such as Oldham and Rochdale. Cob is largely concentrated in the Midlands around Nottinghamshire. Batch is used in two very small areas: Liverpool, in the North West, and Coventry, in the West Midlands. Bap is fairly widespread, but is most concentrated in Staffordshire, the West Midlands (Stoke-on-Trent, Birmingham), and North Wales, Northern Ireland, and the Isle of Man. Bun, similarly, is fairly widespread, but maintains a stronghold in a broad area of the North East, extending from north of Newcastle down to northern Lincolnshire, tracing a diagonal line north of Leeds over to Cumbria. Finally, roll is apparently the normative choice, the most chosen variant and the one with the widest spread across the country, predominating in the South of England and in Scotland. The general picture is of considerable lexical diversity in the North and Midlands, and much more homogeneity in the South.
