Explanation in jtbell linked post looks very nice but I will try to add some of my thoughts.
When you derive HUP you base it on some relation between two observed quantities. Say you can express x using p or something like that. In relativistic case relation will not be linear because relativistic mass (or relativistic Doppler shift) will play a role.
So it seems to me that deviations in relativistic case will be asymptotically bound and that's all.
The most correct way to test this would be to think of some realistic experiment and see how the measurements are taken and interpreted (kind of empirical approach).