When I moved to the UK, it was to be with my now-DH/then-fiance and it was just me moving (no kids) so my situation was a bit different.
We'd thought about applying around March, decided in May to apply for the visa, actually applied in June and was approved in July.
Our visa wait was relatively short, but still so stressful. We did tell some immediate family, but we kept even our engagement secret until we were approved.
It probably was a bit of a shock for people to hear, all in one go, that we were engaged and I was moving to the UK and I was leaving in two weeks, but we decided our sanity was important and didn't was to be nagged with the stressful questions of if I'd heard back on my visa. It's stressful enough just waiting.
For us, it was the right decision to wait to tell people. I didn't have work (well, I did have a few short-term jobs that ended before my planned move date) to organise or anything to pack up besides a few suitcases so that made it easier.
There's nothing wrong with keeping it private for a bit if that's what you want but there's also nothing wrong with wanting to share your news.