Edinburg or London First

As to airport, I don't think it makes much difference. Edinburgh is much smaller and easier to get downtown to Edinburgh. Last Fall we made that run into Edinburgh, train to London, and then we went on to France. Given your choices I would be included to do Edinburgh first.

Unless it is on the Circle line (yellow) - center part of London - everything else will be at least one transfer. Transfers are not a big deal - just follow the signage - it is very good. Mayfair, Westminster, and South Kensington are all reasonably close to the Circle line but being close to the Circle line would not be my first criteria. I am not familiar with either neighborhood so cannot make any recommendations concerning the neighborhoods. We always stay in the Victoria Station area along Belgrave road. Victoria station is principle transit hub and handy for all transportation.

London is huge so being able to walk to sites is not completely practical. So no one place will allow you to walk to all sites.

Since you know you'll be going to both cities, why not let cost decide it? What's the cost of a round trip to London vs Edinburgh? What about flying to one and returning from the other? A multi-city ticket could include an Edinburgh-London flight, or vice versa. (Don't rule out flying vs train thinking it will be more costly. It might not be.)

You can't be centrally located among 10 million people in London. Better to focus on a hotel that's an easy walk to a Tube station. Google maps, etc., will show you what's near the hotel. I try to avoid hotels on busy streets due to noise at night.

London trains to Edinburgh terminate at Waverly Station, in the town center and within walking distance of a number of hotels. A hill or two might be involved, though, so a taxi seems justified.

If you choose to fly into Edinburg first, there is a bus that takes you from the airport to the center of town, Waverly Bridge. It is the Air Link, https://lothianbuses.co.uk/timetables-and-maps/airlink, runs about $6 or 4.50 in pounds. Train transportation to and from London/Edinburg you can book through Virgin Trains, https://www.virgintrainseastcoast.com. I am looking into a similar trip, 3-nights in Edinburg then train down to London for 5-nights.
Enjoy your trip.

"Another concern is suggestions on which underground station is most convenient to attractions without having to transfer lines."

You can pretty much forget about that. There's no such thing as one tube line that takes you to every tourist sight.
It's not that difficult to change lines if you need to.
There are usually tube employees right inside the entrance to the tube who are happy to give you advice and tell you what line to change to for what attraction.
For example, if you enter Tower Hill tube station (near Tower of London), the booth to your left may have a tube employee standing inside it or beside it.
The tube line that comes the closest to what you are asking is the Circle line; a yellow line on the tube map.
Pick up a tube map at any tube station.
http://content.tfl.gov.uk/large-print-tube-map.pdf

In London consider the city bus routes for transportation. Many bus routes run by most of the tourist destinations.
For example, the Number 15 runs from The Tower of London to St Paul's Cathedral, then on through Trafalgar Square, then back the way it came, same route reversed.
You can jump off of one bus at a key location, and jump on another bus to reach a certain destination.
Study the bus map. Just as easy as the tube, but you're seeing more while you're traveling.My favorite way to travel around London.
http://content.tfl.gov.uk/bus-route-maps/key-bus-routes-in-central-london.pdf

If it were a trip for myself, I'd do the following:
1. Fly into Heathrow.
2. Spend your days in London.
3. Train to York. Spend one night.
4. Train to Edinburgh. Stay several days.
5. Fly home from Edinburgh.

How to choose a hotel.
You are not specific about which tourist sights you want to see.
I sometimes stay near whatever museum I will visit during my trip.
For example, next fall I am going back to London. The focus of my visit will be to see most of the British Museum.
I will stay one block from the British Museum and walk there every day.

Frank's suggestion is a good one.
If you stay near Victoria Station, you are near all sorts of good transportation.
Pubs and restaurants, too. Plus many good hotels.

I would fly into Edinburgh and fly home from London, or vice versa. Just check if there's a price advantage to one or the other. Also check your planned dates and see if either city has some huge event that might impact your visit - like London Marathon or something like that. Otherwise, I don't think it makes any difference which you do first.

Agree with others - London is so big that there's no one place you can stay that is immediately accessible to everywhere you'll want to visit. However, all the areas you've mentioned are convenient, with good transport links to sites that aren't right in your neighborhood.

And Rebecca's right - spend some time studying the bus routes - they are one of my favorite ways of getting around London. They make trips that are a pain on the Tube more direct, plus you have the fun of watching London go by! highly recommend.