Do any airlines do a stopover in India on any Australia to London flights?
You are correct. There is no one airline which would allow you to get from the UK to Australia via India direct. The only way you could do it is to fly to say Dubai with Emirates, fly on to India, back to Dubai and so on to Australia. A messy way of doing it but better than say going to Singapore and flying back to India then Singapore then Australia. You might try the websites for One-world, Star Alliance or Sky Team to see if you can get a ticket through them. For example BA to India and JAL or Qantas to Australia. A round the world itinerary may be more useful here. As an alternative to India and bearing in mind you have three weeks, have you thought of going to South Africa? South African Airways are an airline with an excellent reputation and they would fly you from Heathrow to Australia via South Africa. I have been to SA twice and thoroughly enjoyed my time there. We started from Cape Town and drove up the Garden Route which is beautiful and takes in the main wine growing region