We know that as of 2366 in the alternate timeline the Federation is facing defeat within 6 months and that 40 billion people have died. Some other observations from within the episode suggest:
- The Enterprise-D despite being a "warship" is wandering around hostile space without an escort. When we're later treated to Galaxy-class ships in a warship role (in DS9), they were evidently part of larger formations of ships called "wings" akin to task groups or squadrons. The fact that the E-D is alone might give some indication of just how badly the war is going for the Federation.
- Although referred to as a "warship", the Enterprise-D still looks quite similar to the one in the "proper" timeline. Although she has a substantially different bridge module, the basic structure of the ship is the same. This suggests the war started while the Galaxy-class design project was pretty far along, perhaps not as long as the "20 years of war" figure bandied about in the episode.
- Yar notes that the Enterprise has been in service for around 4 years at that point, which might mean that she was rushed into service with minimal internal fittings. (Like described of other Galaxy-class ships in the DS9TM). The fact that they seem to have reduced power to give to things like replicators suggests she's at least a much more spartanly equipped vessel.
Some additional extrapolations from official sources:
- The total casualty figures are surprisingly close to the Borg invasion of 2381. 63 billion vs 40 billion isn't a rounding error, but it is similar enough to give some sense of just how devastating the war has been on the Federation. Given that it's unlikely the 20 year war would've included massive bombardments of several key Federation worlds in the same way the Borg hit out, the casualties were probably far more spread out and suffered in a war of attrition or attacks on civilian infrastructure.
- Klingons may or may not have wiped out the Betazoids in this timeline.
- Since this predates the creation of some powers like the Cardassians, they aren't explicitly mentioned, but given the Federation was in active conflict with both them and the Tzenkethi in the 2350s, it's likely that they were also fighting the Federation along with the Klingons.
It seems likely to me, that the decline of the Federation in this timeline probably stems from having to fight a multi-front war while facing a power with rough parity like the Klingons. It seems unlikely that simply going to war with the Klingons would put the Federation so badly on the back foot, especially since (although some 2 generations earlier) the military brass in the 2290s were confident the Federation could "clean the chronometers" of the Klingon Empire and the weakness of the Klingons in general around the Praxis incident.