I don't think it's fully possible to imagine.
I agree with the earlier posters who (IMO) correctly stated that the influence of D&D was so profound, on so many area, in the 70s and 80s for what came to be known as nerd culture, that it the alternative history is more of an exercise in imagination than actuality (what would have happened if the Persians had conquered the Greeks?).
There are two basic ways to look at the issue:
1. The inevitable/essentialist outlook. The existence or non-existence of D&D at a particular moment in time is of no consequence. Even if there was no Arneson, even if there was no Gygax, even if there was no concentration of wargamers ready to roleplay, a D&D-like product was inevitable in that timeframe. The tides of history would have ensured it. Therefore, sometime around then, a polymath would have created Galleons and Goblins, and we would all be enjoying some fifth version of G&G now.
2. The Great Gygax theory. There was no D&D. The magic never happened. It was the path not taken. Arneson moved on to other things. Gygax decided to keep wargaming as a side hobby and concentrate on "real life" to raise money for his family. RPGs (and D&D) as we know it didn't exist. There was the 70s-inspired mini-fantasy boom with Tolkien and some heavy metal, but it never really went anywhere.
That means that the nascent computer-gaming industry didn't produce classics like Zork, and we lost out on a lot of great text-based adventure games. Not to mention the whole genre of Wizardry, Ultima, Bard's Tale, etc. The ideas of "leveling" and resource management ... even ELF NEEDS FOOD, BADLY never came around. There were never MUDs, or Warcraft, or MMORPGs from Warcraft. But that's not all- a lot of TRPGS and adjacent games (such as Car Wars/Autoduel) influences other CRPGs as well. The whole computer game industry might be a lot more, well, simulationist (flight simulator, anyone?).
That's just for starters; a lot of creatives grew up playing these games and were influenced by them. Would we even have the aborted Firefly?
Heck, Tom Hanks might never have had a career without Mazes and Monsters.
I'm sure that this wouldn't be bad. But it would be very different.
More importantly, it would be that much harder to determine what a D&D coming out today would like. It wouldn't have "modern" mechanics, because we wouldn't have had decades of trying things out. However ... it also wouldn't have the strong 70s "weird/Tolkien" zeitgeist and wargamer roots.
If I had to speculate, I'd say that it would have a lot of ad hoc and bespoke mechanics, like the original game ... but it would draw inspiration from things that are more popular today. Maybe include more non-Western influences and anime/manga fantasy? Heck, maybe the first Monster Manual would have thinly-veiled references to Pokémon.