Suppose you had an "analog" quantum computer, where a register would store a wavefunction $\psi(x)$ where $x$ is a continuous variable (with $0\leq x\leq 1$, say). Instead of gates, you would somehow apply unitary operators to the wavefunction. In this context, it seems to me that the QFT would essentially be equivalent to changing to the momentum basis, $\psi(k)$. This could be done by using a different physical apparatus to measure the state of the register, and wouldn't require the actual application of any unitary operators to the wavefunction. Could there be a similar result for a qubit register which could be measured in two different non-commuting bases? If not, why?
Yes, you can in principle do any measurement as a direct physical operation. The real obstacle is to come up with a way of actually doing it in practice.
The basis states of QFT-ish measurement span every qubit in the register. This means that whatever physical mechanism you are using must be merging together information as large as the register. But physics is local, and your measurement is not, implying you need to build up your big information-merging measurement out of many local interactions. That set of local interactions will probably be isomorphic to quantum gates.
For example, you can do a QFT-ish measurement on photons by putting a diffraction grating between them and the detectors... but isn't putting down the grating just a ultimately roundabout way of applying a gate before measuring?