The answer is probably not. A post on a Mudcat Cafe bulletin board says:
Most English bagpipe designs are based on very scanty evidence such as old pictures and carvings found in churches, or drawings in old books. Occasionally actual relics of pipe chanters are found, which makers study in great detail and take loads of measurements. The rest is imagination combined with experimentation to get a pipe that is playable and in some recognisable pitch.
The Leicestershire pipes were based on such a design that happened to be found somewhere in Leicestershire. There is no evidence that similar pipes were widely popular in that region, but it's as good a name as any to distinguish it from others.But the good news is that I have found a video of Julian Goodacre, who makes the instrument today, playing the Leicestershire smallpipes.
And I still like to think that, centuries ago, this would have been a common sound in Melton Mowbray and Woodhouse Eaves.