I have taken poster tubes from the US to both EDI and to DXB but that was 10 years ago. Fortunately for my Hong Kong trip in 2016 we sent them electronically.
I am seeing far fewer poster tubes these days on flights so perhaps the company or conference you work for can upgrade so that posters can be sent via an electronic format in the future?
I am in the medical field and this has been so helpful to be able to send our posters to conferences this way.
Never ever did I have a problem with poster tubes on any US domestic or US to international flights.
Just don't leave your poster tube behind in the overhead bin or front closet when you exit your aircraft(s)! Yes, I have seen orphan poster tubes several times on my travels.
Have a great and a successful presentation. Enjoy Edinburgh! It is a really great city!