In my experience, the first airline you check in with will set the baggage allowance for the entire direction of your trip, and the allowance is based on the final destination point.

So in this case that would be SN and since you are flying long-haul, you should be entitled to 2 pieces free of charge on the outbound.

I am a little confused as to how many tickets you have though, because you mention flights 1 & 2 are on LH ticket stock, flight 3 is on AC ticket stock, and all the return flights are on UN ticket stock.

If the first 2 flights are on 1 ticket and the 3rd flight is on another ticket, then it will depend on SN's interline policy if they will be willing to check your luggage all the way to your final destination on the outbound journey. If they won't do that, then after your 2nd flight you will need to re-check your luggage and then you will now be subject to AC's limit of 1 free.

Return journey will be 1 piece based on your info.