Very interesting. I've read your other thread and it does seem to be a spontaneous change of name. Sometimes this is due to the person having been given a stepfathers name, then reverting to their birth surname later in life. However that doesn't seem to be the case.
James Adam. Baptized Nantwich 8 August 1844. Parents Mark and Elizabeth. He's with his parents in 1851 and they are living together, but not married, in 1841. Mark Adam, born Scotland, and Elizabeth Higgins.
So either James Adam, the tailor, died or disappeared, and Mary took up with a James Bell, the bill poster, who was the same age and born in the same place as James Adam, or there was a spontaneous name change.
Its possible they were hiding, so either bankruptcy, or owing money, or for a criminal reason.
There is an article in the Manchester Times, Sat Jan 16, 1875. "Salford Quarter Sessions" - Six months : James Bell, Bill Poster, age 29, for obtaining by false pretences at Oldham, on the 14th December, a pair of boots and a horse clipping machine.
I'd be tempted to get a copy of his son John William Bell birth cert. You know from census that he was born in Leeds, and from his baptism in Manchester, that he was born 1st April 1873. So this is probably his birth - John William Bell, Jun qtr 1873, Leeds 9b 462. See what it says about his mothers maiden name, fathers occupation.
Alan.