Everything. When we are talking about flight dynamics and stability in particular, it is the balance of forces or moments that we must consider. For simplicity of analysis, we often choose reference points that allow us to isolate some effects and make them independent of some variables, but if you are interested in the "physics" of it, then you should keep in mind that, in general, the forces and moments are everywhere, and they change.
Specifically, you seem to think that the tail load is zero in the trimmed condition. Not true. It could be true only for aircraft with neutral static longitudinal stability, which normally wouldn't display phugoid oscillations.
In the trimmed conditions, both the tail and the wing create moments (with respect to CG), but they balance out to zero. The essence of static stability is that upon the disturbance, the balance is offset such that it leads to restoration of the initial conditions.
In pure phugoid motion, the upset is airspeed but not angle of attack (AoA). Now, an airplane may have a tendency to have a pitch moment directly in response to speed, which we can designate $M^V$, or better as a coefficient $c_m^V$. This can be related to the nature of its thrust and/or moments from other parts. But this is not strictly required, and we can consider the simplest classical system of just a wing and a fixed tailplane. This system will not see a balance change simply due to airspeed. But consider what will happen:
- Say, airspeed is suddently reduced due to wind shear. This is the only change, and AoA remains truly the same.
- The moment balance remains the same, becuse lift of the wing and the tailplane changes proportionally. Thus no pitch change yet.
- Lift will drop and will not sustain level flight anymore.
- The aircraft will start accelerating down (while still in the same attitude!)
- This acceleration has an effect of increasing AoA (due to the vertical component of speed). Notice that this angular effect depends on the absolute forward airspeed, which suggests that the effect will be lower at higher speeds.
- Now the normal short-period static stability (the $c_m^{\alpha}$ or $c_m^{c_L}$ derivative) will cause a nose-down moment (if the system was stable) and will restore the original trimmed AoA.
- This restoration effect normally happens much quicker than the speed change, especially for larger airplanes. So with a good approximation you could analyse just speed changes and assume AoA = const throughout the process. But physically this is not true, which, I guess, is the source of your confusion.
- With the nose down, the aircraft will gain airspeed.
- QED: the aircraft tries to negate the initial airspeed upset, which is the definition of static stability.
- However, what happens next can be interesting. Due to high inertia, the aircraft will inevitably gain more airspeed than needed, and the process will reverse. Oscillations will ensue, and this is what is usually called "phugoid motion". The dynamics of this motion depends on several factors (absolute speed was already mentioned). Typically, this motion is much less dampened than the short-period AoA stability, and can even be dynamically unstable (i.e. have increasing amplitude). But this is not necessarily a problem: because the motion is slow, pilots can control it.
If you want derivatives, they have already been mentioned. Omitting derivation and with some simplifications, they combine to
$$\sigma_V = c_m^{c_L} - \frac{V}{2c_L} c_m^V$$
This $\sigma_V$ is the measure of static stability by airspeed. It is negative for stable aircraft. As can be seen, aircraft without direct speed-moment response ($c_m^V \approx 0$ - think gliders, for example) will have speed stability fully determined by the short-period AoA stability.
