The cheapest way might be www.megabus.com
Maybe www.nationalexpress…. for comparison
www.eurostar.com to France

My suggestion would be bus in UK (see Don’s links).  Cheap flight from UK to France — use www.whichbudget.com to see which budget airlines fly which routes and decide which UK city to fly from and whether to fly to Paris or Normandy based on the cheapest flight you find. 

I’d take the train for your relatively short trip in France.  You can price train tickets at www.sncf.com

Check Don’s Eurostar link to see if you can find a cheap leisure fare on the day you want to travel — by the time you add the cost of getting two and from the airports, you might find a fare that’s as cheap as flying.

Yeah, it doesn’t get any cheaper than the Megabus in the UK.  Just make sure to book well in advance if you want to get the best price. I bet you could actually get a pretty good deal on the train from Glasgow-Edinburgh and Manchester-Liverpool if you book in advance. pretty short trips.

route planning in france: viamichelin or www.mappy.fr

To travel between London and Paris, you either fly, take the eurostar, or the old-fashioned connections via Dover or Folkstone—- and I think the “old-fashioned” connections would be better for you, since you want to see Normandy.
It’s definitely slower to take a ferry or Hovercraft over, but I think it’s more in line with the experience that you have in mind. It’s interesting to see the White Cliffs of Dover receding into the mist as you cross the channel, and to appreciate that that was the last view of England that many Allied troops had as they sailed to their destinies in Normandy.
I’ll be able to post more details fairly soon, but one possible arrival point is Calais, and from, there, you can easily get to the Normandy beaches.

Basie I think the Hovercrafts have all be retired now, they were on a good day faster than the tunnel and a hell of a lot more exciting.
 
Since you are traveling down from Scotland to england why not pop into Blackpool, its a interesting an very funny place to go, it’s so English it couldn’t be anywhere else.

Hey thanks everyone very much, your advice has been a great help.
Matt