You are right to be skeptical. The presentation by Robert J Dalby misrepresents the findings of Dr. Edson.
The effect being described is not an increase in engine thrust per se, but a decrease in drag coefficient by increased pressure on the missile's "boat tail" i.e. the aft portions of the missile body excluding the nozzle exit. Dr. Edson conflates a reduction in base drag and an "increased component of thrust" frequently in his report, but he clearly understands the difference, and that chamber pressure is not being affected. This phrasing likely caused the confusion.
The phenomena being described occurs in the pressure field of the subsonic, turbulent region between the supersonic air stream and the supersonic jet. Here is a figure from the report showing this region.
The dashed line labeled "turbulence" indicates the boundary layer created by the missile body and the solid line spanning the labels $\Delta Pf_1$ and $\Delta Pf_2$ is the shear layer of the rocket exhaust jet. The subsonic air-flow contained between these boundaries (also bound by the missile's boat tail) is what determines the base drag, with higher pressures in this region producing lower base drag.
According to Dr. Edson's description of the phenomenon is that as the plume expands, the outflow corridor for this flow is restricted thus increasing pressure in the entire region. Note that this does not directly relate to the turbopump steam exhaust but relates generally to an expanding exhaust cone like one sees at low ambient pressures regardless of pump exhaust.
Dr. Edson writes (emphasis mine):
The German workers conclude that the effect of the jet on missile drag is due to jet-induced pressure changes on the boat tail stern of the missile. At subsonic missile velocities the jet exerts an 'aspirator' effect, entraining air in its boundary and accelerating the flow of air around the boat tail, causing a pressure drop there. This causes an increase in drag. At supersonic missile velocities the presence of the expanding jet cone is observed to increase the thickness
of the turbulent layer over the boat tail. The supersonic flow, which controls the effective pressure on the boat tail under these circumstances, is deflected thru less than the boat tail angle or even diverges from the missile axis. The corresponding increase in boat tail pressure decreases the drag, or can even add a net thrust when the supersonic flow diverges from the flow axis, as is at times the case for flow around the V-2 in the plane of the steam exhausts.
The region I previously described in the figure is the "turbulent layer" and it is sketched in the "increased thickness" state described in the first bolded statement. At lower flow velocities the dashed "turbulence" line would be closer (even contoured/attached) to the "missile skin".
The second bolded statement is an example of the conflation I described. Dr. Edson writes about the reduced base drag and the increase in net thrust as if they are two unique contributions. They are not. Any thrust created by the stream of escaping subsonic gas in this region would be a result of the pressure field in the region. That is to say that these two force terms are equal: the latter being a momentum balance expression of the former.
Now they also describe how the interaction of the turbopump exhaust and the rocket exhaust creates an oval shaped exhaust plume. Specifically how the combustion of the turbopump exhaust and the alcohol cooling film expands the plume even further than their mixing alone:
The intense burning of alcohol and oxygen at the point of collision of jet and airstream builds up pressure and acts like a dynamic "plug", raising the pressure In the "lake" of turbulent flow above lt.
So by further expanding the exhaust plume, the outflow corridor for the "turbulent region" is restricted even more and increases the pressure on the boat tail, reducing the base drag beyond a case without this secondary combustion. I take issue though with the bolded statement, as this combustion would surely expand the jet, but would not increase the pressure here, similar to how the combustion in a rocket or jet engine does not increase the pressure. This requires restricted volume (e.g. piston engine) or detonation. Though I concede it could be detonating, but the primary contributor to the reduction in base drag here is the expanding plume.
So to summarize, it is the interaction of the supersonic airstream passing around the missile and the rocket exhaust plume that determines the pressure acting on the missile's base. At low ambient pressures, this plume is able to expand and constrict the air-stream, increasing pressure on the base and subsequently reducing drag. This corresponds with an increase in net thrust. The secondary burning expands the plume even more, increasing these effects. The failing of Robert Dalby and others is assuming that this pressure propagates within the rocket's jet and into the chamber; this is impossible in a supersonic jet. The "squeezing" analogy is somewhat accurate but only in application to the subsonic regions between the airstream and jet. The failing of Dr. Edson is his liberal use of the term "thrust".
I'll finally say that, while Dr. Edson's description of the effects and mechanisms are generally accurate, I don't see in his report many of the equations and flow structures associated with modern studies of supersonic jet/shock/boundary layer interactions. This is likely because when he wrote this report (1949) much of this type of research had yet to be conducted.