You can go online and purchase a ticket from Southampton to Edinburgh as one ticket, having just looked, by default they allow c. 1h to 1h15m between arriving in Waterloo from Southampton and departing Kings Cross from Edinburgh. If purchased this way, I'm 99% certain you are entitled to use the ticket on a later train from London to Edinburgh if you missed the connection due to a late train from Southampton. You could purchase separate tickets with more of a gap between them for a more relaxed transfer, say 2 hours gap, but then if the Southampton train was very late and you missed even that later train, the ticket wouldn't be automatically valid as not purchased as one ticket (of course if your flight was delayed and you missed the train completely then I guess your travel insurance would kick in).
You can buy tickets on Trainline as mentioned which is one of the most well known sites, however each of the half dozen or so TOC (train operating companies) in the UK sell tickets for all train journeys for each others trains, at the same price. I normally use Crosscountry Trains
https://www.crosscountrytrains.co.uk/ as they do not charge a booking fee (unlike Trainline) plus if you buy an Advance Ticket (for a specific train) and then cannot make that train for any reason, you are able to change the ticket for another train on that same journey at a different time (and different day if required) and just pay the difference in fare. The other option is to buy a fully flexible ticket which can be used on any train however these are much more expensive. With the ability to swap the Advance ticket for a different train relatively easily, I find that a much cheaper way to achieve flexibility should my travel plans change.
I don't live in London but would agree with the guys advice above, I'd definitely not be getting on the tube with all my luggage, take a black cab (or Uber/Bolt if you have the app).
My conclusion would probably be to buy a separate advance ticket for each of the legs with a comfortable time gap in between arriving at Waterloo and departing Kings Cross, with the knowledge you could change the ticket for the Edinburgh leg if severely delayed.
One other thing to note is that there are much fewer return ticket options available on UK rail routes that there used to be, it is more normal to purchase a single ticket for each direction of the journey. Admittedly I'm not sure if return tickets are available for your route.
Hopefully the above hasn't confused your decision, please let me know if any other questions ('m no travel agent but do travel multiple times by rail per month down to England with work!!).