If you follow the chain of apparently duplicate questions starting with "Neither is" or "neither are", you will find general agreement that Neither...nor takes the singular verb.
Neither Billy nor Suzy looks like...
However none of the other questions address what reflexive pronoun to use.
Because the verb is singular, and the reflexive pronoun needs to refer to the subject of the verb, that should be singular too. So it's either himself or herself. The pedantic answer is that it's himself because a generic masculine term includes the feminine.
However, themselves has become a generic sexless singular reflexive pronoun, and it's doubly convenient here because we are talking about two people who share the characteristic of a bad photograph.
Neither Billy nor Suzy looks like themselves.
On the other hand, because the sentence is so short, using a singular verb in this case with themselves seems really odd. The effect is intensified if you remove "like" to say that Billy didn't "look himself" (ie., "quite as he normally does," or even, "ill") in the photo.
Neither Billy nor Suzy look like themselves.
Neither Billy nor Suzy look themselves in that photo.
The end result is that you end up either pedant or miscreant. I'd go for the former, but I can understand that the latter is attractive.
Neither Billy nor Suzy looks like himself in that photo.
Neither Billy nor Suzy look like themselves in that photo.