I'm fairly sure the Tilbury thing is a red herring. The image of the register entry is on Ancestry. Alfred lists his father Frank, deceased. There's no mention of Tilburys anywhere in the entry for his marriage. There's no other next of kin entry on a register like this anyway, so not sure where the "He gives his next of kin as a cousin Mr or Mrs, it’s not clear, Tilbury of Silver Town" comes from.
However, the previous marriage in the register is between Harry Handley and Amy Tilbury
I think someone has mixed the two separate marriages entries up