There's no single "best" month — there's a best month for what you want
Apricot season, wine season, and quiet-path season don't all line up.
If you want the apricots
Come in the harvest window itself: roughly early-to-mid July through early August, timed loosely around the Wachau Marillenfest. This is also peak summer heat on an exposed, mostly shadeless riverside path — start early in the day if you're riding through it.
If you want the vineyards and the Heurigen at their liveliest
Wine taverns rotate which grower is pouring week to week through the growing season, so there's no single "wine month" — but harvest (typically September into October) is when the valley feels most obviously agricultural, with picking happening on the terraces you're riding past.
If you want a quieter path
April, May, and the second half of October see noticeably fewer cycle-tourists and river cruise passengers than the July–August peak, while the villages are still fully open and the weather is generally rideable. Spring adds fresh green on the terraces and the tail end of fruit-tree blossom; late October adds autumn colour on the vine leaves.
If you want reliable path conditions
The Danube Cycle Path through the Wachau is a fair-weather route in the sense that matters practically: it's not gritted or specially maintained for winter riding, and river-level flooding, while uncommon on any given trip, can close sections of the south-bank path after heavy rain. April through October is the season the whole regional cycling infrastructure — rentals, luggage transfer, most Heurigen — is actually built around.