PAG - you have the wrong Gillingham. He was Gillingham in Kent, not Dorset.
Gillingham in Dorset did have a framebuilder back in the 1890's, or thereabouts. I think the surname was White and he was in the Peacemarsh area. His claims to fame were having ridden for GB and supplying Dorset police with bicycles. The only reference i have found was in a local history book that i borrowed a few years ago and memory is a bit hazy.
Locally; there was another framebuilder. Used an italian sounding name for his frames and i think he was 753 certified. I kind of remember somebody having one for sale here and maybe even a thread. Again; memory is a bit hazy. I remember it being a bit like Atala, but not them as they actually are italian.
Another local was Hammoon. I think he might have been in Marnhull, or maybe the chap above was. Anyway; this Hammoon is not the present Hammoon Cycles that can be found in Shaftesbury.
Not in Dorset, but Was Peter Hare (Twyford, nr Reading) not a classic frame builder, too? I was sure he built some of his own but maybe he rebadged others. Maybe Dave Russell frames?
Also....up north there was Ottadini back in the 80's.
I am looking at the map on a very small screen so might have missed some of the above. Pretty surre i am right about PAG though.