You are at Heathrow, so even on separate tickets the best option is to fly to Edinburgh.

From landing at LHR it will be a minimum of 8 hours before you reach Edinburgh if you take the train allowing an hour or so for "things".

Even with the hassle of flying and allowing LOTS of time between flights you would be in Edinburgh 3-4 hours before the train - plus you might strike lucky and not have to collect and recheck luggage at Heathrow.

Makes more sense to take the train back to London from Edinburgh, then after your stay take the Eurostar to Paris - under 3 hours even allowing time to check in - which means you'd be in a Paris street cafe before you left Heathrow