UK and Belgium 20 days with 2 kids ( 6 years old and 7 years old girls)
 

Any suggestions for apartments and how can I get the lowest rate for trains?

Plans :

7 days in London, 4 days in Lake districts, 4 Scotland, 4-5 days Belgium/ Holland?

Any advice?

What time of year?

What is your nightly budget for lodgings?

How will you be traveling - car or train?

Our trip will start in late July and end in mid August. Our budget is around 200 US per night.

Plan to take the train for the trip in the UK.

seat61.com gives good train
You know this is high season right, and London in high season is going to sting?
You know two of your objectives are countries? Where in each one would you like to visit.

I'm not sure exactly what you want to know here. Your UK itinerary is fine and not rushed and I assume you'll fly from either Edinburgh or Glasgow to Belgium or Holland?

Trains from London to The Lake district use the West Coast main line with Virgin Trains and you can book online as soon as the seats are released:

www.nationalrail.co.uk
www.thetrainline.co.uk
www.virgintrains.co.uk

You can use any of the above and check for the cheapest fares. They are cheapest when released (3 months before travel) and get progressively more expensive, so book early.

Information on flying from Edinburgh can be found here:

http://www.edinburghairport.com/flig...s-and-airlines

Choose your airline and destination and then click on the logo to go to the airline website. Also try www.skyscanner.net for comparisons.

Thanks for your advice. We finally fix the itinerary. I will get the train from Scotland to Lake District. Any suggestion if I want to hire a car when I arrive at Oxenhomle ?

If you want to rent a car you'd probably do better to take the train to Penrith -- there is an Enterprise location and a local garage that also rents cars.