Holiday cancellation - who gets the APD

You are correct that the payment of the APD (Air Passenger Duty) will only be made to HMRC if you actually travel and that in the case outlined above it would appear that Emirates will pocket it.
Clearly from the amount of APD paid by the two of you, you were not travelling Economy class. I am surprised that Emirates are adopting this customer unfriendly attitude to their for Business/First Class passengers. Have you actually spoken to Emirates about this?
What is the basis of the claim by Saga that they 'don't refund this anymore'? I find this to be a rather unsatisfactory response, more to be expected from the likes of Ryanair than from a reputable tour operator to their First or Business class customers. How far in advance of departure did you cancel? I assume that you have checked that actual Emirates (electronic) tickets have in fact been issued and have the Emirates (electronic) ticket numbers. I am asking this to be clear that Saga have in fact paid Emirates and that it is not Saga who are pocketing the APD rather than Emirates.
Why have the insurance company excluded it from your refund payment? I would have thought that the insurance company should have paid out on the entire claim then attempted to recover the APD from Emirates themselves.

Emirates say that Saga should contact them to process the refund.

Yes that makes sense. As Emirates were paid by Saga they should normally refund to Saga who should in turn refund you.

Saga say that they can’t do that.

Why not? Neither the refusal nor the apparent lack of cooperation makes any sense.

ABTA will only be able to assist in respect of problems with their member, SAGA. Emirates are a scheduled airline and nothing to do with ABTA. My view here is that your problem is with SAGA not Emirates who appear to be acting correctly in referring you back to SAGA.

I still remain puzzled as to why the insurance company have chosen to deduct only the APD from your refund and not for example any other taxes and charges which may have been included in the cost of the ticket. Did they state a reason for their decision to withhold payment of the APD amount?
Another question which comes to mind is if SAGA are refusing to contact Emirates, have the tickets ever actually been cancelled as obviously some contact between SAGA and Emirates would have been necessary to do this.

Could you clarify/confirm that what you purchased from SAGA was an inclusive tour (package holiday) and you paid one single payment (or deposit then balance subsequently) to SAGA to cover the entire holiday cost including flights, accommodation etc.

I would suggest a small claims court action against SAGA.