Having just done this a few times - Google Maps always routed me through Heathrow Express as the fastest option. Typically 1 hour from London Bridge area to Terminal 5.

From MK: take train to Euston, walk to Euston Sq, take Circle/Met/Hammersmith & City to Paddington and onto Heathrow Express.

If it's running through by the time of your trip then I'd just get on Elizabeth Line from Euston, probably via Northern Line to Tottenham Court Road.