I don't have any idea how big one such government could get eventually.
But the first practical steps, for a 1-jump distance, might look like this:
A vote shall be scheduled for date X. All votes shall be cast on that day on all worlds.
Votes from other worlds shall be recorded and sent by express jump to Capital.
Votes shall be tallied on Capital; this process may not begin until all worlds' data packets have arrived.
In effect, you vote on the 1st, the votes travel and are counted on 8th or 9th, results announced on Capital on the 10th, and word spreads back to you by 17th or 18th.
If you are willing to wait several months before anybody knows how things turned out, the government can get bigger and bigger on the map.
Of course there are problems - use IRL for inspiration - which I can't solve ahead of time from here. (Alas.)