I believe that the Weimar government would eventually built up the armed forces wether they would start a war depends on who's in power at the time.
Without the Nazis what would a rebuilt German military look like in the 40's-50's?
I assume this would be a von Schleicher succeeds scenario, where he becomes Germany's strong man instead of Hitler. Weimar was pretty much dead by 1932, it was a question of who was going to be the dictator. Hitler was the dictator of the disaffected lower middle class (among others), while Schleicher was the military dictator. So Germany under him would see many of the same goals of Hitler: casting off Versailles, stopping loan/reparation payments, remilitarizing the Rheinland, probably annexing Austria, potentially grabbing the Sudetenland, and of course revising the Polish border and annexing Danzig.
However Schleicher wasn't looking for war like Hitler and probably would have taken longer to achieve those goals, but would have followed the same path of military build up similar to the Nazis, but would likely have stretched it out over a longer, more sustainable timeframe.
The players would have stayed the same probably, but would not have been replaced with Nazis. The one difference that I can think of would have been von Blomberg, who was Schleicher's political enemy, which would have meant von Hammerstein-Equord would be the Defense Minister, not Blomberg (who was actually exiled to a frontier post in Prussia by Schleicher in the 1920's). Otherwise its likely that Wever would have the Luftwaffe, Raeder the Navy, potentially Milch in the Aviation Ministry, and definitely Schacht in charge of rearmament and the Reichsbank.
Goering would not be around obviously, nor the Nazi military force, the SS, but we would see the Gestapo taking out Schleicher's enemies, as he was in charge of the Black Reichswehr in the 1920's (they carried out assassinations of Germans cooperating with the Allies' inspectors).
The German military would be unified in a single command (OKW) and rearmament carefully carried out under the army's aegis. The Kriegsmarine would be shorted in the build up, with the Luftwaffe still getting a reasonable share of the funding pie, and the army the lion's share.
As to composition, I think by the 1940's and 1950's it would look like the 1939 force that Germany was putting together, but much better trained and younger (obviously) than the jumble of trained soldiers, conscripts, and WW1 veterans drafted in 1939.
Overall the force would be extremely professional, like how von Blomberg was trying to expand the army slowly to maintain quality IOTL before Hitler screwed up his plans, but smaller than the force Hitler demanded. I expect the military would not be ready until 1945 to actually fight a war, due to a slower, less expensive build up, and the need to maintain quality by extensively training the new conscript classes. Also the need to build up trained reserves would mean that for years training would be needed to have an effective reserve in case of war.
The equipment would be high quality, mirroring the German propensity to over engineer like in WW2. When all is done, I think the German army would be about the size of the French army, potentially bigger with reservists, but that would depend on the state of the German economy.
As a side note I don't think the Soviets would have as much of a build up as in OTL, because without Hitler shaking up the world order in Europe by threatening war constantly, which Schleicher was too cautious to do, as well as German military build up being slower than IOTL, there wouldn't be the same tensions. Germany would be bigger and economically more powerful in the 1940's than in 1932 that's for sure. Schleicher would expand, but keep it reasonable to avoid war. Even with just Austria Germany would be quite a bit more economically more sound/powerful. I do wonder if Schleicher would renew ties with the Soviets that Hitler severed in 1933. Of course the cooperation with the Reichswehr and Red Army had withered by 1933 without Hitler anyway, but perhaps a new, less ideological leader would have looked for allies in the East to counter the 'Little Entente'?
Edit:
The Luftwaffe would certainly have had a strategic bomber. Britain isn't the target though, but France, Poland, and potentially the USSR (Ural bomber?). I don't see the failure of the Do19 or Ju89 getting funding in this budget environment, so we would likely see a He177 without diving requirements (Wever's death likely butterflied and no way Udet would be in the Luftwaffe without Goering), though the twin engine disaster might still exist, meaning the He177B (four prop) still shows up in the mid 1940's.
Without Goering the Luftwaffe is technically and tactically far more proficient. It isn't sattled with the terrible intelligence arm of OTL, nor with a series of terrible leaders. Much more professional over all.
Since this thread I recently posted was ignored, but is very relevant here:
https://www.alternatehistory.com/discussion/showthread.php?t=258481
The German military would be far better educated on strategic issues and issues of war economics. Overall the senior leadership would be much better trained by the 1940's-1950's. The Wehrmacht (Reichswehr?) would be a far more competent force at every level than IOTL.