There are several problems with such a question, and this certainly qualifies as a "When did you stop beating your wife?"-type of question. Why? Because it assumes, a priori, that photons CAN have speeds other than c in vacuum. This is not verified. So the idea that a photon gets born with some low speeds is not an idea that has any physical basis, and thus, the starting point is all wrong.
Secondly, there is a problem in reconciling our experimental evidence with such a scenario. Let's look at this carefully.
Say we have a body that is initially at rest. at some point, emits a particle, as shown in the figure below.
Now, let's say that the smaller body then accelerates, by some means, to some final velocity, as shown below.
However, this final velocity v_f has no directly relations to V, i.e. it isn't correlated to V since the conservation of momentum of the two bodies no longer is relevant here. v_f no longer carries any direct information about V.
So let's look at what we know about such a thing. Atomic recoil, electron recoil, and a while bunch of other experiments on photo emission and photon collision experiments have shown that what we measure in such interactions totally conserve momentum. In other words, we measure v_f (since it was already at c for photon), and this v_f is still correlated to V via a direct conservation of momentum. This clearly means that v_f is equal to v_i, and therefore, there is no "acceleration" of photons
This scenario applies to a whole zoo of fundamental particles as well since the same conservation law applies to many such interactions involving these particles.