Nuclear weapons are inevitable, yes. They are implicit in the physical laws that govern nuclear reactions, so if those are uncovered then there's no stopping the development of nuclear weapons eventually. Getting the power of tens of thousands of tonnes of HE into a projectile you can shoot from a cannon is very attractive militarily. But it's not inevitable that they would be developed then, or first. Absent Big Mistake #2, I think it's more likely that nuclear power plants and propulsion systems would be developed first (perhaps by the 1950s or 60s), with nuclear weapons some time afterwards.
It's also worth considering that ICBMs and their nuclear payloads are a 1950s solution to a 1940s problem ("how do we destroy an enemy state's industrial base?"). If that problem does not arise, then the solution to it may not arise either, and the form that nuclear weapons take may not look the same as IOTL. For example, if there's no perceived need to destroy industrial facilities from the other side of the globe, then there may be a much greater emphasis on tactical nuclear weapons, with yields down to the tens of tonnes, for use on "conventional" battlefields.
Now, some of the logic governing the use of nuclear weapons will remain the same. An enemy airfield is a legitimate military target, so a "tactical" warhead can be used against it, and of course you need some way of delivering those warheads, so a long-range rocket might make sense. Once the nuke arrives at the enemy airfield and detonates, it will not restrict it's effects to the airfield perimeter, so the city nearby will be affected, and once civilians are being attacked by an enemy then there's great pressure to attack enemy civilians in retaliation. We saw exactly the same thing with aerial bombing raids in WW2. I think that there would be a creeping process towards the types of weapons and delivery systems we have in OTL. It would probably be slower, though, and influenced heavily by the problems (and solutions) that were uncovered in TTL.