A question on the Gillette family

This post has nothing to do with Provence but it's French-related. Like me - I think. Here's the question.
My mother's grandmother was a Gillet, pronounced with a soft g, French-style.
She lived in the north-west of England and we know nothing much about her family. My mother was always given to understand though that she was of French extraction, and that her family were Gillettes, and I wonder if anyone knows more about the name's history?
There are a zillion genealogy sites of course but most relate to the US and not England / the UK. The only notable Gillette I've come across was the bizarrely-named King Camp Gillette - he of the disposable razor - whose family was ruined by the Great Chicago Fire.
I found out that he set up factories for the razors in England in the 19th century so possibly he sent a male relative to the North West.
Gillette was a bit of a nutcase as far as I can see. He was a Utopian Socialist, which is pleasant if, well, utopian. (We've all been there.)
He wrote a work called The Human Drift arguing that the US should form a single gigantic industry owned by the public and that Americans should live in a single vast city, Metropolis, with electricity generated by Niagara Falls.
Hmm. Nice.
To follow up, he wrote World Corporation - a programme for creating this US Mammoth Corp. - which sounds a bit like a....job application. Anyway, having had his Big Idea - the ra-zor - he obviously became a Big Thinker. Or perhaps he just took his first name very seriously. He offered to pay Theodore Roosevelt a million dollars to be World Corp. president. But Roosevelt was already, um, American president and he refused the offer. (Probably rolled his eyes and said to an aide: "Do me a favour. Write some sort of polite Not-on-your-life, go-away-and-stop-hassling-me note to that weirdo Gillette. Him and his goddammed World Corp....")
Anyhoo - the business was eventually sold to Procter and Gamble, in 2005, where his razor joined all those other things you use in the kitchen and bathroom. But Gillette died long before, in the 30s. He'd lost a packet during the Depression but was still rich enough to frequent the Desert Inn in Palm Springs, turning into one of those anonymous squillionaires who wanders around in a tatty old dressing gown. The owner, asked how come this vagrant was tolerated, replied: "...that's King Camp Gillette. He has practically kept this place in the black the last few years."

Apart from the vast fortune, it sounds horribly like he could be one of my forebears. But, how would I find out?