Matty --- There are 2 birth registrations for a Roger McNutt Bateson -- first Q/E Jun 1903 -- second one Q/E Dec 1904. (First one died in 1903)
According to Lancs BMD these births were in the sub registration District of Darwen --- mother's maiden name -- Bateson.
If you look you will probably be able to find Roger Bateson, born about 1904 -- with his mother Fanny on the 1911 census --- (we are not allowed to give details from this census!)
In 1901 a Fanny F Bateson aged 32, born Everton -- is living in Ellison Fold, Darwen, shown as a Widow.
It would be useful to obtain Roger McNutt Bateson's birth certificate maybe -- it may just give an address which could be checked to see if John was there.
regards
June