Some servers restart automatically as soon as it gets to the year 2050, so by figuring out how long a game usually lasts and when the current game started you can make a guess as to when the next one will start.

The servers run by the OpenTTD reddit even have an online status website with game logs (https://ttdredd.it/). I tended to play there as whenever a game started, there were always a half-dozen players online.