train from Edinburgh to London

You will need to use the Tube. Either buy a 1- single ticket (most expensive if you are staying in London), 2-an Oyster card and load it with pay-as-you-go money or a single, 3 day or 7 day travel card, or 3- a paper 7 day travel card (can be used for the 2-for-1 deals). Then take the Piccadilly line from Kings Cross to Leicester Square and walk to Charring Cross from there. It should take just under 30 min.
( I think that is the quickest way, but you could check out alternatives on the Transport for London website). You could also use the buses, but I am not familiar with the routes.

The Journey Planner on the Transport for London site suggests taking the Victoria Line from King's Cross/St Pancras to Oxford Circus and the Bakerloo Line from there to Charing Cross. The trip will take 10 minutes. 3-day travelcards have been phased out. Your choices are 1-day and 7-day travelcards. The cash fare for a single ticket is £4.50.

Please don't try to walk from Leicester Square to Charing Cross with heavy luggage, it's a very crowded area. If you are willing to take a longer period of time but not have to change trains, you can get the eastbound Circle Line around from Kings Cross to Embankment station, which genuinely is within an easy walk of Charing Cross. (The station is at the far end of the Charing Cross platforms, although you'll then have to walk to the other end of Charing Cross station to go in.)

I wonder if the usually reliable TfL Journey Planner has sprung a gasket and panicked on this trip. I generally allow, fairly accurately, 3 minutes per stop plus another 3 to 5 for each connection. It is 5 stops via Oxford Circus plus the change. Don't be tempted to take the Victoria Line and change at either Euston or Warren Street onto the Northern Line Charing Cross Branch unless the screens show a problem on the Bakerloo. The connection, as I remember it, at Oxford Street is a simple cross platform (and through a short tunnel) connection; at Euston it is a bit of a walk as it also is at Warren Street. What looks shorter may actually take longer. There isn't a reasonable bus connection between the two stations.

I revisited the TFL Journey Planner and found that I gave a false report. The estimated ride time ranged from 12 to 14 minutes.

No one has mentioned taking a taxi. That's what I would do if I arrived at King's Cross with more than one small piece of luggage and/or I was tired. Taxis cost more than the Tube and might take longer depending on traffic, but it's a tradeoff for convenience. Hauling multiple pieces of luggage around on the Tube is not fun.

Thank you all. Appreciate the advice ! Sarah

If you book the Advance to Charring Cross rather than to Kings Cross it will cost £3 extra but include the cross-London transfer. A bit more than the Oyster fare, but a bit less than the separate cash fare. Depends really on whether you need an Oyster card or want not to have to pick one up at Kings X.