Barbara and Alan Pidcock write:

"There is a record of a Gilbert Pyttcocke in Cambridge in 1298 and another Pydcock in York in 1301 (no christian name).  At that time both these places were big trading centres dealing in exporting raw wool to the Continent, and there are records of Cambridge Markets in the 13th century being full of Flemish Merchants trading in wool.

If Gilbert was a Flemish merchant it would explain a few things, for example the scarcity of Pidcocks in England, and we could also explain away the lack of growth in the family between 1300 and 1500 by the Black Death which wiped out a third of the population followed by other plagues and starvation.  They were difficult times.  It is possible I can get more information on this as merchants had to be licensed by the ancient Manor Courts, and there may be records still extant.

Another more pie in the sky assumption is that the College of Heraldry says that 'cock' as a suffix to a name means 'son of' and that Pidcock could be son of Peter.  This didn't make much sense until I realised that the Flemish equivalent of Peter is Piet or Pieter.  Maybe Pieter was the first merchant from Flanders leading to Gilbert Pyttcocke.  It's just a thought, but if right your DNA testing might take you further than England. "

This idea, that the Pidcocks are descended from an early "incomer" from the Continent, either part of an invading army or a merchant is congruent with some early Pidcock given names. Henry, Anthony, Gilbert and Vincent do not appear to be early English names.