Sorry, no sign of Peter or just "P" Mylechr*st or Mylechreest that I could find in Findmypast The Military Collection:
"....allows you to access over 38 military and service record datasets, spanning a period from 1656 to 2005 - the most comprehensive set of British military records available online. It covers the two world wars, as well conflicts such as Waterloo, Trafalgar and the Boer war, and peacetime army lists. You’ll find officers and soldiers of all ranks, and other servicemen such as RAF, Navy, Marines, and support personnel..."

There were a few other Mylechreest/Mylechrist men, but not P.
Any other spellings? (Need the first three letters before a wild card.)

Sue