Detailed Instructions: Given a travel related situation, and given a specific question about the situation, summarize the question into a title
Problem:I am travelling from Prague > London > Dubai > Sydney.
The Prague to London is with BA and London to Dubai is with emirates.

They are booked on separate tickets so I don't think my bags will be checked all the way through. Does this mean I will need to collect my bags and pass through customs and immigration at Heathrow and check-in again for my next flight? Or will BA do it for me?

If I have to, how long does this take at Heathrow? My flight arrives at 12:30 PM and I am non-EU passport holder.

My connecting flight with emirates leaves at 2:15 PM so I only have 1hr 45min to do this assuming the first flight is on time.

Is this possible? Or should I try to make other plans?
Is there a way to fast track immigration or customs? I am a One World Sapphire member.

Solution:
Do I clear immigration and customs in heathrow T3 for stop over on seperate ticket