Dear Charlotte, Couple of things (and the old letters you talk about might shed light) I got a William Gudg christened in Preston in 1803 on the FamilySearch site - I can't think he could have really been anything other than a badly-written or transcribed Gudge - and that does keep the family in the North West.
One of the websites gives access to all UK phone book details. Basically there seem to be four main concentrations of Gudges (makes us sound like an endemic disease!)
London. I'm told by my aunt - Dad's sole surviving sibling that when contacted someone in that family denied any link with the Liverpool Gudges; obviously don't know how true that is.
Birmingham. Me, my mum and my brother - obviously link with Liverpool as we know.
You and your direct family in Liverpool.
The South West. These seem to trace back to a Samuel Gudge about 1600 in Dorset but you also find the name in Devon, Hampshire, Somerset, Avon and Cornwall.
Don't know if any of the Gudges from that corner have the world know of a family link between North West and South West - mining perhaps?
Final bit. Some of the Gudges outside Birmingham/Liverpool may be my Uncle Paul's kids or grandchildren; he had 7 children and adopted 2.