As a non-Brit, I find Brits brag regularly about their children, but nearly always with the caveat that they put little effort in whatever achievement and done in a way that it's something every child should be able to do whatever in the right circumstances. Something unique or requiring effort is not for British discussions, on either side of that spectrum.
When my kids were younger, I heard variations of 'oh, I didn't do anything to help them learn to read, they just started at [early age] from having books around' or 'they just started doing multiplication tables, it's amazing how children can pick things up'. Talking about putting effort to helping children learn was seen as hothousing and that a child worked hard at a subject or area was just not the done thing. Thankfully, they're now all of an age where I rarely have to talk to other parents about what they're doing though sadly it took far too long for me to realize most of those parents were lying, it took it's toll to feel like the only parent who had to teach their children these basics.
My parents were terrible for the 'what my children did well is because of us and what they do poorly is through their bad choices that they should be mocked for'. Between that and the above parents, I tend to not talk about what my children do with others as I don't want to get into it with others or have my children feel I'm making it all about me, so when I do I mainly talk about their accomplishments to my children about it and whether to them or others, I focus on their effort and the challenges they've overcome. The things I would have liked acknowledged as a child I want my children to hear me talk about and celebrate. I'm not really concerned if it seems to come out of no where to other people, but more how my children feel about how I'm talking about their accomplishments.
And yes, I've had my children surprise me and had little-to-no idea before they've earned something. My oldest has been particularly good for this - he has a significant stammer that has been a source of frustration for him - him getting into public speaking to the point of being rewarded by his college for it was a surprise. I would not have guessed how much that would excite him to get into until I saw him doing it. Him getting into face-to-face volunteering that he earned a 100 hour award at 15 surprised me - hell, it surprised him, neither of us had been great at keeping track and he'd just done all the hours he could. He went out in all-weather, dealt with all sorts of people who reacted to him in very different ways, did the training courses required, he worked hard to be a help. I don't think most people around us would have had much of an idea of how much work he put into that as it's not something either of us talk about. Not sure if he's very British in that or not.