Two singular noun phrases connected by or which together function as the subject require a singular verb:
(1) [My auntie or my uncle] is arriving by train today.
Please note that it is always the first verb (is in this case) in the verb phrase functioning as predicate verb (is arriving in this case) that agrees with the subject.
Also note that in the English examples that are used to illustrate these rules about subject-verb agreement, as well as in the examples in the box above, the relevant subjects appear within square brackets, while the heads of the relevant subject noun phrases and the first verb (i.e. the agreeing verb) of the verb phrase appear in boldface.
Finally note that since my auntie and my uncle is a conjoined noun phrase, we have two noun phrase heads, namely auntie and uncle.