Having now found out further information, it seems the dead relative was a male cousin of my mother's. She never mentioned him, to my knowledge, although did mention his father, her uncle - however, at the time I was unsure who he actually was (I was a child and only met the uncle once, I think, this was about 50 years ago!). This cousin was apparently unmarried and had no children - or certainly no surviving children anyway. My elder bro and I have been doing a bit of research and have discovered that our maternal grandmother actually had 3 siblings, we think - the dead cousin is the only descendant of those siblings we have found any mention of, however. The heir hunter firm say they have not managed to trace any other living relatives, other than myself, my siblings and our 6 cousins and one uncle (we're a relatively small family). My mother had 2 sisters and one brother - all the sisters are now dead, Uncle H being the sole survivor of his family. As such, and from what I've read, he is actually the main descendant, and therefore priority beneficiary, of the dead relative. So it's quite likely he is the one who will inherit whatever estate the dead relative left.
Unfortunately, Uncle H is in his mid-eighties and has advanced Parkinson's disease, however, he does have a limited memory of his dead cousin, he says they didn't have much face to face contact but mostly exchanged Christmas cards. He reckons the cousin didn't even own his house, as far as he can remember, so unless the cousin was a miser who sat on his money and never spent anything, there won't be much of an estate to distribute anyway.
So none of us cousins are expecting to become millionaires, or even receive any money whatsoever actually, as we're not the primary beneficiaries. It's been interesting finding out some family history we knew nothing about, though.
It was tricky enough finding out information about our relatively small family - how husband could ever manage (if he was so inclined, which he's not really) to trace all his family history I don't know, his family is huge. E.g. I have just 6 cousins and 4 nieces/nephews, whereas husband has around 40 nieces and nephews, and countless cousins, some of whom he's never met.