As previously discussed on ELU (Impinge: transitive or intransitive?, “Get on”: is it transitive, intransitive or both?), many sources reserve the term "transitive" for verbs that take a direct object, and define "direct object" as something like a noun phrase that is a direct complement of the verb. This definition of "transitive" excludes verb constructions involving a prepositional phrase ("prepositional verbs"), where the direct complement of the verb is the prepositional phrase (which in turn contains the noun phrase as the complement of the preposition).
The sentence "I wondered when you'd get around to telling me that" has no direct object: it takes the prepositional phrase "to telling me that" as its complement (where "telling me that" is the complement of the preposition to). (Technically, in some theories, "telling me that" might not qualify as a direct object in any circumstances because it is a clause rather than a noun phrase, but that's a different issue. For the purposes of your question, we could avoid that topic by altering the example to something like "I wondered when you'd get around to it" or "I wondered when you'd get around to the point of your visit").
The linked Wikipedia article "Phrasal verb" discusses the (possible) distinction between "phrasal verb" constructions involving particles and "prepositional verb" constructions involving prepositional phrases. The expression "get around to" makes use of both the particle "around" and the preposition "to".
Some particle verbs take a direct object, and so would be considered transitive by all terminology systems. Wikipedia gives the example "You can take on Susan"/
"You can take Susan on", where "Susan" is a direct object (the two possibile word orders indicate that on Susan is not a prepositional phrase in the first sentence).