The press secretary announced that neither himself nor the

C as well.

The reflexive pronouns such as himself indicate that the sentence subject also receives the action of the verb. Personal pronoun He is more approprisate here.

go with D. They later in the sentence needs a plural subject.

I think I got it wrong in my earlier posting because I was confused about the use of 'they'

Here is the explanation:

The pronoun does not act like a verb, in that it does not need "and" to COMBINE two singular nouns/pronouns to form a plural verb.
If I simply say, "Jane went to the store. Susan went
also. They met at the store" you can see that the pronoun "they" refers
to both girls although they are in entirely separate sentences. This
cannot happen with 2 subjects and a verb.

So saying 'neither he nor the President would be available for questions until they had had more time to examine the report '
is entirely accurate.

So C it is
This question is from Barrons and key says C too. I see no explanation for the answer though.
Hope my answer is convincing enough.