Looking at a selection of things, a pattern of sorts appears to exist.
His father was chief engraver of the seals and a medallist, but only appointed after his son. It has to be borne in mind that coin output at the Royal Mint was negligible over much of the reign of George III. Coppers from 1797 and 5 shilling dollars were produced at the Soho mint, leaving the Royal Mint to produce mostly gold, so a full time engraver was unlikely to be necessary. Appointed probationary engraver in 1811, his father had not yet been appointed chief engraver of seals and William was not yet on board, so it would appear that a simple W was adequate - such as on the Military Guinea. The question arises as to whether any others were employed as engravers prior to 1816. I don't know the answer to that.
Where T Wyon Junior has engraved a medal, he signs it T Wyon Jun, and conversely his father uses T Wyon Sen. Pistrucci was appointed on Junior's death and used anything from BP to his full name. J B Merlen, engaged from 1820, used his initials, as did WW (employed from 1816). At this point the use of a single W would be ambiguous due to the appointment of Thomas Wyon Senior, leading I believe to the different identifying initials