I checked my previous comment regarding when the audio is actually loaded. Please keep in mind I’m not a web developer, and I am just guessing from the web developer console that Safari shows me.
During review the audio is not pre-loaded. Only when you get a reading-answer right, it is loaded.
I suppose that is the reason why the audio is delayed for people with slow internet connection.
But here the interesting thing: During lessons, the audio is pre-loaded. Before you reach the ‘Reading’-tab, the audio is already loaded at ‘Kanji Composition’ and therefore you experience no delay when the audio should be played.
However, as soon as you get into the lesson-quiz, which has a similar code-base as the review-session, the audio again is only loaded when you get the answer right, thus you experience a delay again.
The best thing would be that the audio is loaded as soon as you reach the reading-question of a certain review item. So that when you get the answer right, it is immediately played (assuming that you cannot read+understand+write the answer all within 1 second 
).
Someone could probably even write a userscript to pre-load the audio 
Or we just wait for WaniKani staff to fix it.