As to whether it is non-zero, but small enough to neglect anyway in some larger calculation: impossible for third parties to say. How small it is depends on how large [$]T[$] is in my simpler example, and so probably how large [$]T' - T[$] is in the real example, as well as the other details of that case like the unspecified [$]a(x_u,u)[$] and drift. Once those unspecified things are given, a simple Monte Carlo could answer the question: "how small is it relative to other quantities of the same dimensions in the larger calculation?"
The context of the problem is the following: for [$] T'-T [$] small enough we can write
[$] C(S_T, T, S_T, T', \sigma_T) \approx 0.4 S_T \sigma_T \sqrt{T'-T} [$]
For [$] T' - T [$] not so small, and choosing a slightly different forward strike, I can derive
[$] \text{Something I can observe today} \approx E_t \left[ S_T E_T \left [ \sqrt{\int_T^{T'} \sigma^2_u du} \right ] \right ] [$]
Of course I can always work under the share measure, and then I'm done, i.e. I can relate the expected forward realized volatility ( the forward start volswap strike) to something I can observe today, but under the *share measure*. The problem is, I am not sure how to interpret the expected forward start realized volatility under the share measure other than that the drift is different. Maybe I should look at the variance strike under the share measure to gain more feeling for it? But, if I insist on working under the risk-neutral measure, the independence question above pops up.
However, as in general there is no independence I think focusing on what expected future vol under the share measure means is maybe more useful.