I tried a new Allen key from a set that Heather bought for her push bike and the reluctant bolt came out and was replaced. So the purpose bought tool was never used.
My family have given me a new pushbike for my big birthday and when I tried to raise the seat the metric Allen key seemed like it would round out the little bolt. I'm sure everything on the bike would be metric, but I tried an imperial hex key and it fitted much better.
So for future reference, chaps, consider trying an imperial hex key if the metric key is a sloppy fit. That will save "rounding out" the "hole".