As others have noted, Buddhism was known in the Greco-Roman world but struggled to attract adherents. This may have been partly due to trade intermediaries meaning that contact and opportunities for conversion were more limited.
But it may also be that Buddhism was far enough outside of the Greco-Roman conceptual universe that it was a difficult sell as a conversion attempt. My impression has been that early Buddhism didn't really have much success outside of regions which shared a similar conceptual universe (roughly the Indian subcontinent and some nearby areas influenced by cultural contact).
This changed with the development of Mahayana Buddhism, which seems to be more successful at spreading to areas outside of that culutural sphere (Central Asia, China, Japan etc).
The development and spread of Mahayana Buddhism probably came too late to spread west as effectively, since by then the Abrahamic religions were fairly entrenched.
But what if a different school of Buddhism emerged earlier, which wasn't tied to the original conceptual universe? Not exactly Mahayana, but something which has some of the same features.
In terms of timing and synergy, something could arise within the early Epicurean philosophers, if one of them met with some of Ashoka's missionaries and had some fresh inspiration. Perhaps Hermarchus, as the leading Epicurean after the death of the founder, had some ideas...
I've toyed with this idea in the past, but never tried to turn it into a TL.