When you have a series joined disjunctively by the word or, the number of the verb is determined by the number of the noun closest to the verb, that is, the last in the series.
One apple, one orange, or two bananas are then added to the blender.
Two bananas, one apple, or one orange is then added to the blender.
Check out this embarrassment:
[W]henever the president or the governor issue [issues] an executive order . . . .
Bob Dunn, Contemplating Our Future, Tex. B.J., May 1992, p. 448 (quoted in Garner Legal, p. 841).
As you'll see in chapter 6 of the eBook Understanding the Parts of Speech, this rule also applies to the either . . . or and neither . . . nor correlative conjunctions. Thus, number is governed by the number of the noun closer to the verb. Look at these examples:
Neither the coach nor the players want to lose.
Neither the players nor the coach wants to lose.