I'd take the car. No doubt about it, especially with the children.
The train will be a load of hassle, especially with luggage and connections. The car won't have that. Plus you can take additional luggage specifically to amuse/occupy the children that will be too much to fit on the train.
With the train, once on it you're trapped (which I hate) whereas in the car you can stop and stretch your legs at any point. We often stop for a picnic if we see a pretty spot, or divert from the main road into a pretty village if the mood takes us.
Manchester - Dover is a longish trip, but if the children were happy (I only transport neices/nephews, don't have my own) then we'd do that in one hit just stopping to change driver halfway (it's about . Then use the hour and a bit in the ferry for a meal and decent break. (When I've driven that trip in the past it's just over 4 hours, and I have been known to drive it solo, so not too bad).
I'd also be tempted to do the final six hours from Calais to Limoges on the same journey. BUT! IMPORTANT! do NOT use GPS/routing advice and go through Paris! You'll lose too much time. Go via Abbeville/Evreux, Chartres before picking up the E05/E09 route. That also has the advantage of not being on the toll roads, and has pretty good scenery.
Yes it'll be a long day and going from my experiences in the past I'd start in the early hours of the morning and literally just get the children out of bed and into the car without waking them if possible, but it has the advantage of having the fortnight at the destination rather than splitting the journey down into 2 or 3 days (which in my experience is just as tiring).
Overall in the car I'd allow about 4.5 hours for Manchester - Dover, 2 hours for rest and Channel crossing on Ferry (this includes the booking in time and early arrival) and then 6 hours in Forn Parts. So that's a long 12 hour day - what's the train time door to dor?