score:4
Oneworld has recently changed policy and airlines including BA are no longer required to interline bags on separate tickets. In your case, Emirates isn't in oneworld and it's likely that BA wouldn't have checked-through your luggage anyway.
You will have to exit customs and immigration, collect your bags, change terminals (if necessary) and check-in again. Given 1:45 connection time, this is very risky and I personally would not attempt it.
If you could travel with hand baggage only, you might have a chance of making the connection, but it would still have a risk, and leave you with the possibility of having to buy a new ticket if your incoming connection is late.
As a oneworld sapphire, this does not give you any special fasttrack immigration at LHR for short haul flights. There are kiosks for machine readable EU passport holders which it seeems you are not eligible.