Best way to go from France to Beaune-Burgundy region

Have you checked flights London to Lyon against the cost of Eurostar & TGV?

You should be able to get a Prems fare to Dijon for 15 euro pp one way, if you time it right, there are even some 9 euro fares. Use the TGV pricing calendar.
http://calendar.en.voyages-sncf.com/en/
Beaune is just a short hop away on frequent regional trains.
You'll need to get to Gare de Lyon from Gare du Nord on the RER D.

One other thought , to consider ; Once you are in Burgundy , how will you get around ? Any number of places : Guedelon , Semur en Auxois , Montbard ( Abbey of Fontenay ) and many more pretty much call for a car to get around the general area . I would consider that as a possibility .

Assume you mean London to Burgundy.

Yes, rail your best and fastest bet unless you want this trip to take all day. If you need a car (a good idea), rent one there.

You need a car in Burgundy and with 4 people it gets cost effective. Any chance you could take a low cost flight to somewhere in Burgundy and rent a car there? Or pick up a car in Paris and drive to Burgundy. Once in Burgundy you need a car to see most of what is interesting. WE usually pick up a car at Orly airport when heading for Burgundy.

You can also think about the Eurostar to Lille and drive with the rental car all the way to Dijon. Shortest route is the E17 tollroad via Reims, some 500km so you can avoid Paris. Fuel will cost you about €60, the toll will be around €35. About the rental of the car I think it doesn’t make much of a difference if you have it earlier in the day picking up in Lille or later that day picking up somewhere else. So if you can get for instance 4 Eurostar tickets for €50 each, the total budget for going this way from London to Burgundy will be some €300.

From Nord, you can drive to Beaune or Dijon in less than four hours easily.