Many thanks for all the ideas coming in so far.
I have searched FreeBMD extensively in a similar manner to Toni. There are several possible births for Harry, but without some census records that might help to give a positive link to his parents they must remain conjecture. The other possibilty is to find a marriage certificate for Harry and Jane. As well as looking for them in FreeBMD I have been through all the marriage index lists from 1883 to 1888, (their first son was born about 1887) looking for a marriage of a Kimb*, Harry, Henry or anything else that looked possible, without success.
Note also that in 1901 Jane claims to be aged 40 and born in Bromsgrove, but there are no appropriate births in Bromsgrove, although there are a few in neighbouring registration districts such as Kings Norton and Stourbridge.
Finally there are no appropriate entries in the IGI that I can find.
The Horsfall link looks hopeful, but there are not many of this name in the records in the area we are searching. Don't forget Harry is believed to have been born in Birmingham, but that could, of course, include Aston or Kings Norton. In fact Selly Oak, where the family are living in 1901 was in the Kings Norton RD.