I assume this is for your Europe and UK trip. I think if you fly a major airline with larger planes (not a regional contractor to the major with small planes), and if your layovers have adequate time for the big box to get from one plane to the next, you should be in good shape. Even better if you can do it without a layover so there is less opportunity for mis-handling in an airport.

Smaller short haul planes on rare occasion have too much luggage to be able to get everything onto a plane, the biggest box might be the first thing that gets left behind.