I do not profess to know what the law says in detail.
I do not profess to know what the law fully means.
I do however know about a case of a PPL going from CAA to FAA and still flying G Reg aircraft.
He was allowed to fly G Reg Aircraft by the CAA, he was allowed to fly them in UK airspace, but was instructed by the CAA to contact each European Authority in which country he wishes to fly a G Reg aircraft.
He did this, for example France responded in a few weeks, and he always flies with a copy of the letter for the authorities at that particular airfield.
Never under estimate the power of people of authority on an airfield; if they are not satisfied that you can legally fly the G Reg aircraft, you will not be allowed to fly, regardless of what you think, what you are told on this forum, what you are told by someone in an office and regardless what a lawyer says.
By the time you clear up the problem the aircraft could have been impounded for a few months.
If there is doubt there are guaranteed problems, because the people on the ground have the same questions as you do; but they can stop you.
Get a letter from each country you will want to fly to, it is not too much of a problem. Or, convert to a JAA licence then this is purely accademic.
Good Luck
FB