Without an external reason for why missiles don't exist, this is unrealistic
No tactical nuclear weapon has ever been used in a combat situation. (Source)
The only reason to not use strategic nuclear weapons is because you want a "surgical" conflict, you want to preserve something. This has a simple name:
Leon Trotsky once made a very practical observation:
There are no absolute rules of conduct, either in peace or war.
A war is fought either to gain something or to stop another from gaining something. The question remains, what is a nation willing to risk to gain that something, or stop another from gaining that something?
So far, the world has been willing to not risk everything. But our single data point, our own history, suggests that there is a limit. That limit is the step into tactical nuclear weapons.
Hollywood sometimes sucks
Hollywood is in the business of selling a cheap fix. Quick entertainment. And so they need to present fundamentally broad ideas in very short periods of time. Consequently, Hollywood makes it seem like tactical nukes basically don't exist — that the step into nuclear war is a whole lot shorter than it really is.
Maybe Hollywood's right, but it's probably not. With the exception of lunatics,1 the world has proven in all but two cases2 that it will not take that step to strategic nuclear war without a whole lot of something happening first. Oh, we'll posture and cast the proverbial vulgar hand gesture at one another — test a few nukes in out-of-the-way places to prove we have them — but we won't actually pull the trigger.
But we've never seen what nations will do if someone pulls a tactical trigger.
Tactical nuclear warfare is the trigger for strategic nuclear warfare
I can't think of any reason why anyone would roll out strategic nuclear forces during a conventional war. I'm not even sure I'd believe a story that presented it as a last-ditch effort to win an already lost conventional war. That doesn't reflect the subconscious fear the world feels about nuclear weapons. Likewise, I can't think of a reason why any country capable of a strategic nuclear response would hesitate to use it if faced with a tactical nuclear attack. Why?
Because the doofus is willing to use nukes at all
And your only option is to stop him dead in his tracks before he lobs even one more shell.
Having lived through the cold war, the Cuban missile crisis, and a number of other doomsday clock events, I can tell you that one of the greatest reasons no one would dare use tactical nukes is that the rest of us.. all of the rest of us... are scared to death of it.
So, what conditions might allow tactical but not strategic nuclear warfare?
The "target" does not have strategic nuclear capabilities and isn't important enough for those who have strategic nuclear abilities to stand up and fight back in their name.
Enough time has passed on your world that everyone who remembers the nuclear testing, use of nuclear weapons, and any cold-war-like conditions that threatened nuclear strikes, and at least one full generation beyond that, are dead. (In real life this might, maybe, happen somewhere between 2080-2100.) No one really remembers just how bad the consequences or even the fear were, leaving the world stupid enough to think that a tactical attack really wouldn't be that bad.3
A nation develops what it sincerely believes is tactical nuclear protection for their military, opening the door to the idea of a "safe" tactical nuclear war.
Note: I can't actually imagine a world with artillery but not rocketry — but that's what you'd need to have tactical nuclear capabilities but not strategic capabilities. I consider this incapable of attaining suspension of disbelief, but that's just my opinion. But, I do have one more idea...
- Zugzwang. It's a chess term that describes a point in the game when neither player can make a move without significantly degrading their defense. Tactical nukes might be believable if you set up your story with a condition where neither military can make any move without basically giving up the war. Their only option is to use tactical nukes and hope like crazy the other guy won't respond with strategic nukes. This would actually make a fabulous story (I mean it, a Hollywood-options-paying-for-your-grandkids-college kind of fabulous), if you can seriously and believably set up the zugzwang. In this case, it's a race when the two combatants realize the zugzwang is forming, realizing they can't avoid it, and rolling out the tac-nukes first. You can believe the PEOC discussion would cause heart attacks all the way over in Tibet.
1 By "lunatics" I mean religious fanatics, terrorists, heartbroken nuclear scientists... you know, anybody who has access to a weapon of mass destruction but doesn't have the common sense to not actually use it.
2 Nagasaki & Hiroshima....
3 If you'll forgive the momentary descent into politics, here in the U.S. we're experiencing a resurgence of Socialism despite decades of world experience with Communism. But we've begun to forget what pure Socialism and Communism can make of a society. You may agree with me or not at your leisure, maybe I'm wrong, but my point is that it's simple to do something stupid once enough people have forgotten how stupid it is. Very few people seriously study history, which is why it's a suspension-of-disbelief-worthy justification for the use of tactical nukes.