If he was b. Fulham his birth may be recorded on the parish records of All Saints Fulham, which do not appear to be indexed on the IGI. (They will be at the LMA and possibly the Fulham & Hammersmith local archives will have copies) There's some possibilities in the Fulham 1841 census for his relatives:
Thomas Jenkins, 47, Gardener, born in county
John, 18, lab to ditto, born in county
Mary, 46, born in county
and
John Jenkins, 46, Gardener
Sophia, 46
Frederick, 17
George, 12
Ann, 10
Benjamin, 8
Elizabeth, 6
Emma, 4
I can't spot a Thomas Jenkins, gardener, of the right age, in the 1841 census in Middlesex.