Hi - a good flight case is expensive… but often times a good standard (molded plastic, not wood) case will do fine. It depends on your horn vs your case. Make sure you pack it so it does not move at all, make it as tight in the case as possible. Also, put instructions/pictures inside the case for them to put it back in correctly, should anyone need to remove it. We've all heard Al Baer's story about the upside down horn. Inside the lid I tape both handling and packing instructions as well as a photo of the horn in the case. I also tape down all hardware on the outside of the case that could get caught or broken off with gaffers tape. (I always carry a roll and 'prep' the case when I get to the airport)
A big part of the equation is also overage charges. Fortunately, for most non-USA carriers (avoid United, Delta, etc at
all costs) flying internationally with a tuba is actually EASIER than flying domestically. Most cutoffs are 50 lbs, sometimes 40lbs, and almost all tubas are over the maximum dimension combination for most (but not all) airlines. Also the cutoffs seem to be somewhat flexible, and are enforced by the counterperson at will, where you check in your luggage.
I can't recommend British Airlines highly enough. As far as this writing, and knock on a HUGE piece of wood, they are the
only airline that does not have overage charges for musical instruments. They have a special exception. They've also been super to deal with any time I have had issues.
Check the current details and policies online, but there are a few important caveats I have learned:
1.) Your horn has to be your ONLY checked baggage. Carry everything else on. If you try to check a suitcase in addition, they will charge the **** out of you for your horn. It's a loophole.
2.) Allow at least 3-4 hours of changeover time for Heathrow. You have to manually make sure you open up the layover time when booking the flight. Less than that and I can almost GUARANTEE your horn won't make it to the next flight. When the layover is under 2 hours, pretty much like clockwork, the horn gets delayed by at least one flight. Fortunately, they seem to take pretty good care of it.
Simply book a long layover; there is a good Asian restaurant in Heathrow called Wagamama, sit back and have a leisurely lunch while you allow plenty of time for them to switch your horn from flight to flight with no worries

3.) Do not lock your case, they will break it open.
4.) Don't try to book flights that switch from Heathrow to Gatwick, just don't! Or if you do, give yourself an EIGHT (yes I said eight) hour layover.
5.) Always try to check in at least 2 hours before your flight.
Customs to Western Europe is usually not an issue. To them, and fortunate for you, most European countries are pretty familiar with tubas… and musicians. They are looked at in a much higher light than stateside. There are stories of people with customs issues, but they usually deal with multimillion dollar violins and guitars made of certain contraband woods.
To Canada is a different story…
One more thing, get horn insurance. I use Music Pro and I have had good luck with them. It's not expensive and is worth it for the peace of mind.