There are an awful lot of similarities in what's being offered on the websites. They're both offering the machine I was looking at at the same price - and no other company is. Both are offering a 15 year guarantee, with free servicing up to once a year. Both are offering interest free credit over 3,6,9 or 12 months. Both offer discounts if you phone or chat to them. Both have the same promotional video for the machine that I haven't seen anywhere else. And, as you say, the phone numbers are very similar (only the last two digits are different) - and both are London numbers, even though only the British Sewing Centre is based in London.
There's certainly some sort of connection between them.
The address given for Sewing UK is occupied by an import company based in Japan, but there is no apparent connection between Sewing UK and the import company apart from the address. I can't find Sewing UK on the Companies House website.