there are ways you can narrow this down you know. If she was about your age, then you can look in the general register office index for birth registrations registered in the town/district your uncle lived. (unless you have an idea that this was a relationship conducted beyond his home town)
If the mother registered the baby on her own, then she could not add the father, even if they were co-habiting, unless they were married. So, if you do not find a child registered with his surname, you have a bigger search, but if you can narrow down the district, you can request the register office look at each entry for that year and any that are illegitimate (i'e no father on entry) they can supply copy certificates. Then you need to follow those births through to marriage to wee if any of them added your uncle as their natural father on their marriage certificate. that is one way , probably expensive, but may be worthwhile.