For a lap child, most airlines charge 10% of the fare or 10% of the miles. You will have to check with the airline that you will book the ticket with. Bassinets are available in all classes. I suggest you fly business since you have enough points. Some people will argue that you get the baby his/her own seat. Your choice.

Here are couple of options:

1. Transfer points to American Airlines AAdvantage (yes, it's US based airline but you can book flights on BA and Qantas). Go to AA.com and search one way Aberdeen to Perth using flexible date. You will get a calendar showing availability. Business class is 120,000 AA points for 2 and about $900 in fees. I see plenty of options next April/May. Call AA and add the lap child.

2. This would be my choice as it is most direct on a top airline. Transfer to Singapore Air's KrisFlyer. Go to it's website and search for "redeem miles" one way London to Perth. You will see saver Business award options for 170,000 miles. You get 15% off for booking online so you only need 144,500 miles plus 861GBP. Call SQ and add lap child.

Another option will be to redeem for 3 economy tickets on Singapore for 102,000 miles plus fees. You can then request bulkhead section with bassinet.

For the SQ redemption, you can use Avios to get to London.

No stopovers are permitted on a one way redemptions.