Is there any way you can amend the air ticket to be AUS-LHR-EDI?

4hrs would be about right, but the risk is your flight is an hour late, immigration is backed up and the tube is rubbish, meaning you end up cutting it fine, or otherwise you land 90 minutes early, breeze through immigration have a seamless tube ride in - then spend 4hrs with your bags at Kings Cross station.

One other option might be to buy an Economy Plus fare on BA LHR-EDI. Cheapest I can see for a random date in May is £83. This offers a free same day change if seats are available - https://www.flyertalk.com/forum/brit...change-ba.html.

So book the last flight then just find a ticketing agent at Heathrow & bag a seat on the next flight.