Are English diphthongs nasalized before nasals?

Solution 1:

For the question as to how to represent nasalized diphthongs in IPA, Portuguese offers the answer. This Wikipedia article says:

Portuguese also allows nasal diphthongs that contrast with their oral counterparts, like the pair mau /ˈmaw/ "bad" and mão /ˈmɐ̃w̃/ "hand".

So in Portuguese, you put the nasalization diacritic on both elements of the diphthong.

In comments, Araucaria pointed out that a tilde over both elements of the diphthong: [a͠ɪ], would be better notation for a nasalized [aɪ] in IPA. However, this doesn't seem to be the standard ... possibly because of typogrphical considerations when IPA for Portuguese was first introduced, although it's not hard to do using Unicode today. And if you want an extremely narrow transcription, you might want to let your choice of symbols depend on how far into the diphthong the nasalization starts.