Yes, anyone exporting anywhere needs to follow compliance rules, because the point is to ensure that
all exporters are in compliance.
Consider the OP's example:
> Have a part made in Japan, integrated into a product in the states but sold by someone in the UK to in France? you'll have to comply with Japan, US and UK laws.
The reason all countries tediously ensure their laws are being followed is because, if they weren't, there'd be an obvious legal loophole: you could simply proxy export items to a country with different export restrictions -- and then all export restrictions would be worthless.