I know very little about these, so I can't be of much help;
What do you mean by "speed limit"? If it were a tuned amplifier, it would have to track the frequency the whole way. A self tracking tuned amplifier would be amazing, and such a thing either exists or can be made, but is very slow and ponderously expensive (phase tracking circuits + servo drives + vacuum variable capacitors!). Slow as in, takes seconds to track.
A true wideband amplifier simply amplifies what goes through it, so there is no frequency-time trade-off, it's all frequencies [in the passband], all the time.
Now, I understand a TWT is dispersive (a consequence of the helical transmission line), which means you won't get phase coherence -- a square pulse in will end up as ripples (much the same as ripples on a pond -- gravity waves are dispersive). The frequency domain components will end up amplified more or less correctly (given the peaks and dips characteristic of the amplifier), but the phase shift or time delay will vary significantly with respect to frequency.
If you're looking to do something like wideband pulse or burst RADAR, you'll probably have a hard time. If you're doing chirped, you'll need to calibrate against the chirp characteristic of the tube itself, but given that, it should do just fine.
So, it depends what you want to do with it.
Tim