To answer the original question, given the technology of the day, it was a waste of time and resources to even consider a turbojet engine for a cruise missile like the V-1. It wasn't until long after the war when companies like Williams and Turbomeca produced small, relatively cheap and efficient turbojet engines that were way more practical for cruise missile application.
The idea was there earlier, Like the Fairchild J44 and the Westinghouse J32 ( 9.5 or 9.5A, only 9.5 on on diameter) but getting engines that worked and were cheap enough to be expendable may have been a problem in the early years.
The designs Ohain was working with pre-war aren't all that far from the simpler/smaller range of designs that appeared post-war, even more similar to some of the smallest turbojet designs ever mass produced. (in part due to radial inflow turbines being more practical to use at those scales -they also ended up taking over most automotive turbocharger designs and I do wonder if Hirth considered applying them to their own turbocharger developments)
They were not particularly high tech, refined designs either ... and some of the innovations came less from engineering and more common sense mechanic/technician input. (the first functioning liquid fuel combustion chamber used burners adapted from the design of Otto Han's blow torch, even relying on hydrogen for the warm-up period before gasoline/kerosene could vaporize properly -similar to using methanol/ethanol in the spill pan of a torch -you can use gasoline too, but it's horribly sooty and can foul the fuel jet ... the very problem Ohain ran into when trying to start the engines using gasoline)
Still, I'm slightly surprised they hadn't attempted propane, butane, ether, or alcohol fuels to expand testing beyond hydrogen both for warm-up and for higher power compressor/turbine tests while the combustion chamber was being refined.
Granted, all work was being focused on engines suitable for manned (re-usable) aircraft, not missiles, and there's probably a better argument for using jet powered bombers than cruise missiles. (even with the poor fuel consumption, and fairly large frontal area/thrust ratio, if they could have refined the HeS 3 or HeS 6 for mass production rather than hitting the many many snags it took to getting the HeS 8 even to flying condition, they might have had useful short-life/frequent TBO engines much earlier in the vein of the earliest production 004Bs -albeit you'd need 4 to power a light/medium bomber ... but all the better for limping home in the case of in-flight failure)
I believe I've seen reference to Heinkel targeting a 10 hour TBO/service life as the minimum for practical military use.
Heinkel might have gained more RLM interest with a bomber than a fighter too. (and the way Ohain's engines were designed, they were very well suited to the sheet-metal working equipment of Heinkel's airframe facilities, adding potential for at least limited mass production before the Hirth Acquisition -the latter should have helped improve a lot of the engineering difficulties though, including with the compressor, diffuser, and turbine -or perhaps convinve Ohain to consider a centrifugal compressor + axial turbine layout for larger designs -plenty to go off from Hirth supercharger and turbocharger developments)
I'm off topic again here, though ... and I suppose a dedicated early-war Heinkel Jet Bomber 'what if' deserves its own thread anyway.
I don't see how aerial reconnaissance to do a sort of bomb damage assessment for the V-1 strikes could possibly have worked. Just look at the distribution of the impact points. The 42 V-1s that fell around Southampton were in an area roughly 33 miles by 5 miles! An aircraft making a pass over Portsmouth or Southampton, or flying up the Thames estuary to make a few passes one side or another of the river would be lucky to distinguish any damage caused by the V-1 strikes.
Might using recon flights able to follow/detect the V-1s with radio transponders be more useful? (corroborating the results of the data being sent back)