Thanks. So he was indeed illegitimate. That clarifies things.
In the 1921 census James Fawcett, aged under 1 year, is in the index in Blackburn, West Lothian, registration district 669/2, enumeration district 3 page 6. With the same reference are five more Fawcetts: Marion, 44; Thomas, 22; Robert, 20; Janet, 16; Alexander, 14.
James Fawcett, aged 58, died in Glasgow in 1978, mother's maiden name not listed in index. After 1974 mothers' maiden surnames, if known, are always indexed except if the deceased was illegitimate, so this omission suggests to me that he is likely to be your James Muirhead/Fawcett/Campbell.
As it's less than 50 years ago you will need to order a copy of the death certificate to confirm that it's the right person, or get someone who is going to a Scotland's People Centre to transcribe the details for you.
Incidentally another James Fawcett, aged 1, mother's maiden surname Campbell, died in Blackburn in 1914.