Apologies for confusion, I believe the portrait should be entitled William Traies of Crediton father of William Traies the artist of Crediton. The artist Lemuel Francis Abbott, per Wikipedia, lived from 1760 to 1803 (5/12/1802) and painted notable people of his era Nelson, Herschel, Cowper, Matthew Bolton & John Wilkinson, the industrialists etc. He lived in London and was declared insane in 1798 so would probably not have received many commissions after that. You'd think he would have been expensive to hire. William Traies the artist's dad lived from 1764 - 1849 and was not a rich man. In 1798 would have been only 34. Portrait seems of an older man than that. As William Traies the cordwainer was a wealthy man, born in 1728, and specifically mentioned his portrait in his Will I think it more likely that he is the subject of the portrait. I've been meaning to contact the museum and find out what facts they have got like provenance etc. - the fact they are seeking information suggests they may be themselves unsure. William the cordwainer's brother, John Traies the cordwainer, appointed another London portrait artist, Abraham Wivell, as an executor of his Will so there may be another connection there.
I will revisit the branch of the family you are researching and see what I can turn up. Be in contact again after I've done that.
Good to contact again - many happy memories of this thread - still find useful information on it when I revisit it.
Regards,
Bob