Here is the situation: with 20 seconds left on the clock a player is leading on points at the end of a round.

What do the rules (if any) stipulate on beginning a new round?

I've been on both sides of the issue, needing that extra round or really not wanting to play it. I always follow my personal maxim of don't be a Dbag and put yourself in the other player's shoes but I am curious to any official ruling on this.

As soon as you finish cleanup and start planning, you're in the new round, technically. But the precise point it changes is often pretty ill-defined when there are multiple things to do between the last ship shooting and the planning phase beginning, but not all of them are always necessary, and you can often start some of them before they're due.

Most people I've seen have said 'if a dial has been set we play it out', if there's any real chance of a change in the points. So fling one dial down and take your time setting the rest, then change the first one.

The idea that a dial has to be down for the new round to happen isn't supported by the rules - if you've resolved the End Phase then you're in a new round's Planning Phase already.

End Phase Rules Reference

That said, it's best to communicate with your opponent about pace of play etc. *before* you get to that point so that both players are aware of and agree on when the game will end.

Yeah TO's in the UK I've seen play it as "if a dial has been set, you're into the new round and play it out".

I've seen a person dump a random dial down on a non important ship with seconds on the round left - allowing them to then play out the round, set the crucial ship's dial properly and manage a final round turnaround win.