Hi, Diane. This is just a note that J1c3 is a very old mitochondrial haplogroup. The current Family Tree DNA mtDNA haplotree shows that it has 12 distinct subbranches below it. These would represent newer splits, chronologically, resulting from newer mutations.
J1c3 and its subbranches have been found in ancient remains all throughout Europe, from Neolithic France to the Early Bronze Age British Isles to the Russian Steppes and into Siberia. One paper (Behar, et al., 2012) estimated the time J1c3 first appeared to be within a range of 6,600 and 11,000 years ago. An estimate for its parent haplogroup, J1c, is that it first appeared about 16,500 years ago.
The top-level haplogroup J does show its highest European concentrations today as being in Cornwall (20%) and Wales (15%). But that also reinforces that the haplogroup J1c3 by itself won't give you adequate information on which to base an assumption of genealogical relationship. Someone else who carries J1c3 might not have a common ancestor with you for thousands of years.
Taking a full-sequencing mtDNA test at Family Tree DNA would give you more information, i.e., which subbranch of J1c3 your mitochondrial DNA is plus the capability to match other test takers, but whether it's worth the expense is up to you. Even with a full-sequence test, exact matches might still be separated by as many as 70 generations; at a 50/50 probability of matching, it's around 31 or 32 generations (Andersen and Balding, 2018).
If you encounter someone who does not have "J1c3" at the start of their haplogroup, you can definitely rule out a matrilineal relationship. But even an exact mtDNA match can span too many generations to be very useful as a form of positive evidence without applying a great deal of genealogical research and analysis along with it.
As others have noted, your autosomal DNA is the best place to start with DNA for genealogy. That can give you a lot of valuable information as far back as your 3rd and possibly 4th great-grandparents. And the ability to compare both autosomal DNA and your J1c3 results to the matrilineal connections you find gives an extra level of confidence about the evidence.
It's just that mtDNA is more difficult to work with accurately for genealogy than it may seem. It's a remarkably tiny DNA molecule with very little room to mutate. That's why it isn't very diverse in the world's population. We're expected to reach a global population of 8 billion before the end of this year. There are currently 5,468 mtDNA haplogroups cataloged; that number has been essentially unchanged since 2016. So that means for every mtDNA haplogroup there are over 1.46 million people. The actual ratio is even higher because most people will be identifiable in one of the deeper, newer haplogroups, like the subbranches of your J1c3, rather than the higher, older branches like just J1c or J1.
It's great information to have. I had my mtDNA first tested at a low resolution 20 years ago. Then upgraded the test, then upgraded it again to the full-sequence version. But mtDNA is better at helping you rule out who isn't a match than it is in offering positive evidence about who is a match.
Best of luck in the search!