The only time I've ever come across it was a gym which allowed older teenagers (older than 16) to attend at set times a week and have use of the gym. This was set up because a small group of (mainly) boys, who had trained with the club, had set up an offshoot parcour and free running group and wanted to train in a safe place. It was a trial thing and I'm not sure if it carried on.
Other than that no.
It's quite common for gymnasts to train with another club while they are on holiday. That said, UK gymnasts are covered by BG insurance which is transferable to any UK club. You'd just need to check your own insurance situation, but I know it happens with overseas gymnasts visiting this country, so it must get sorted somehow.