I think Caesar is too smart to think the NCR was able to launch a nuke at the Legion.
And the NCR might think the Legion are too dumb to be able to target a nuclear missile at them.

And LR supposedly takes place right before the battle at Hoover Dam. By that time The Courier has made a name for himself. And everything big that happens in the wastes is most like done by The Courier. Just my two caps.

I suppose that there's one way to look at, "But how did they know it was me?":

The assassination squad found you out there in the wilderness so it's reasonable to assume that an otherwise "invisible" recon team is tagging you and directing the assault squads in. The recon team has overwatch during the assassination team's contact and makes a post action report on the firefight. The same could be true when you launch the missle(s).

The reputation system basically just skips the "word gets around" step because it gets inconvenient. I mean for the Powder Gangers in Goodsprings for example. They know Cobb is there, and eventually someone might check on him. When they arrive though they're just gonna see dead bodies and hear the story of the Courier.

The assassination squad I think is more obvious. If I were to hire an assassin right now to kill an important political figure and the assassin simply never returns, I don't assume "a bear ate him;" I assume he got caught and killed.

As for Lonesome Road, we don't know. We do know however that the battle between the two Couriers under war-torn skies is supposed to become legendary. Even Dog/God hears about it. Hell even CHRISTINE hears about it, and she opts to stand vigilant at the Sierra Madre for eternity.
HOW it became legendary, we don't know, but however it became legendary was supposed to be incredibly broadcasted so that everyone and their mother hears about it.

Possible eye bot? Maybe a transmission sent through the radio airways? Probably a recording or something.

Could you expound on this? Where do they talk about the Courier/Ulysses battle? Did I miss something?

In their endings, if they live.
God/Dog specifically has to get the fusion ending, whereas Christine always hears about it.

Like I said, we don't know the how, but SOMEHOW it becomes legendary.
Perhaps the nuke you set off when you made the Courier's Mile could be seen from further out than we realize? Maybe people simply saw the nuke and word spread, and the few people who would know about Ulysses put two and two together (Johnson Nash) and figure you and him must be behind it.

Is this only if you complete LR before DM? I did DM first. I merged Dog and God and don't remember this, but I am notoriously absent-minded so I may have drifted off.