It depends on the qualification process. If all teams other than the host nation are required to qualify in continental tournaments, the GB squad may as well contest the European qualifiers together.Briefly r/e the GB problem and getting a team into the Olympics - Curling at the winter Olympics has it sorted, the highest placed and qualified UK rink (team) will represent GB, and it's always Scotland of course.
So, in cricket, there could be a home-countries playoff to decide which of the four (or more likely two as NI is "Ireland" I think, and it's the England and Wales Cricket Board so that's covered) represents GB.
Now, whether England would qualify and then draft in some Scots or Welsh lads, let alone Morgan, would remain to be seen.
It would be easy to handle if ICC full member nations were only allowed to enter under 23 players. England could keep playing Tests and ODIs while the GB team (i.e. players from England U23 and Scotland) contested the European qualifiers and the Olympics themselves, with no U23 players in the England team for Tests there's no conflict. The same would apply to Australia's team for Tests.
But given that it is the ECB veto which is keeping the Olympics off the table, by the time it ever gets on the table the Scottish board will either have a much larger say in the matter - either by having a bigger voice at the ICC if the ECB loses its veto or by having their own National Olympic Committee which will have an equal voice to that of the UK.