Scot Rail do not serve the Glasgow to Paddington line.
The company with the fastest service would be East Coast train which run fast frequent trains down the east coast from Edinburgh to London Kings Cross. There are frequent services by Scot Rail between Glasgow and Edinburgh.
There are also the Virgin trains which go between Glasgow and London Euston on the West Coast route via Carlile and the Lake Country and Crewe. These are somewhat slower because the line is not straight and there are many more built up areas along the route.
There is also the Sleeper which goes most nights between both Edinburgh and Glasgow, arriving London Euston around breakfast time.
I'm not giving the name of the operator with the sleeper because it has just changed and I don't know what it will be called, nor do I know yet what changes there will be in the operation. That is the one sleeper I would still ride - it has a fabulous reputation and is very comfy.
The name of the trains between Edinburgh and Glasgow, and throughout all of Scotland may be changing from Scot Rail because the Dutch have just won that franchise away from First Group.
It was First Group that just lost the running of the Sleeper, too.
There will be 2 timetable changes before you travel, in December and May, and with all the chairs moving around the table certainty will be light on the ground. We also have no idea how the new operators will run their operations.
And, to add to the hilarity, there is likely to be a new owner for the East Coast fairly soon.
Paddington from Kings Cross or Euston can be a false economy of time. The Heathrow Express is certainly the most expensive train in the UK per mile, and while frequent it is not as frequent as the Tube and you have to get to Paddington by Tube or cab first.
Kings Cross to Heathrow is about an hour by Piccadilly Line with no changes.
Euston to Heathrow by Tube involves a change, either one stop on either the outbound Victoria Line or one stop on the inbound Northern Line Bank Branch, both of which will take you to Kings Cross, where you change onto the Piccadilly. The ride from Euston to Kings Cross is about 4 minutes. That connection can be made step free at both ends.
Or Euston to Green Park on the inbound Victoria, then onto the Piccadilly Line to Heathrow. That can also be made (nearly) step free.
The advice to check in the near future to see how the land lies is a good one. You will need to be flexible.
The biggest concern I have is for such an early departure from Glasgow on a Sunday. There tends to be engineering work on the lines on weekends, especially in August, and British trains tend to sleep in on Sundays anyway.