Will I miss out on any opportunities by procrastinating?

In Oblivion, there's a clearly defined calendar, with roughly one game minute passing for every two seconds of real time.

My question is this: do any events happen at specific dates and times, and is it possible to miss out on them by side-questing or procrastinating, or do all events occur based on my progress in the game?

I think in the base game there are no specific events.

Some NPC's do move around, and every once in a while some get killed. But most of them do not have that problem. (And I think most of the patches removed the NPC's that just got killed by wandering monsters, or they are just unkillable).

In certain cases this isn't true. The first thief guide quest for example is timed. As you need to race after another potential thief guide recruit. One of the main quests where you need to stop the siege machine is timed. But that again, is while you already started a quest.

Other quests while they seem like they are timed. Like starting the invasion of oblivion, don't do that much. No need to rush to clear all the oblivion gates. The world isn't going to end.

Even more interesting, while games like ultima 7 had quests that you could trigger by just walking though certain areas (or flying over them with the carpet). Oblivion doesn't do this. So no worries. It is rather hard to mess up quests by just loitering a bit.

There are even a few unkillable followers that you can get during certain quests, and by then not completing the quests you can have them following you around forever. (Until you finish their quest).

tl;dr.

Nope. Take your time. Events are quest driven, and not missable.