My Italian friend, Lele, wanted to spend a fortnight touring the UK and Ireland with wife and 2 kids outside of London and I managed to convince him to choose two of

1) Ireland
2) Southern England
3) Wales and the Midlands
4) Northern England
5) Scotland
namely 4 and 5.

Assuming he wants to use low cost airlines I suggested fly into Liverpool or Manchester. Visit Liverpool, Manchester, Chester, Leeds, York, Durham and Edinburgh with a side order of some exciting Scottish hills/landscapes in June 2020 using a car and flying back to Italy from Edinburgh.

What have I missed out and where, given these tight timescales would you go in Scotland?

Assume I have the Yorkshire bits covered.

That is an awful lot of cities. No Lakes? No Dales? Hadrian's Wall? Coast?

Like all good Italians, they want to go to Liverpool for the Beetles and Lele seemed surprised that the UK conducted the slave trade from there . MAN or Liverpool airports makes more sense for a curving route allowing for the best of Yorkshire.

Yes, city-centric it is, for no more reason than it was part of an Italian/English conversation. I have discussed Fountains Abbey a fair bit along with the Wall and the Lakes (he lives next to Lake Como so a bit of a busman's holiday). So filling in the Yorkshire bit will be easy.

Tell me more about " Alnwick and Bamburgh, Edinburgh, and a few days in the Trossachs or in the northeast castle trail"

There are many very cool castles in the Northeast of England between Newcastle and Edinburgh.
Alnwick is famous more recently due to Harry Potter fame. Bamburgh has a spectacular site over a beautiful stretch of beach. Dunstanburgh is a majestic ruin on the coastline. Warkworth is one of my family's favorites, a bit smaller, intact but unfurnished - a lot of fun for children. You could easily see two of those in a day since they are not far apart. With children I think the castles would be a highlight - in summer they sometimes have events there where they offer childrens' activities.. They could also check out the Farne island boat trips from Seahouses, which is on that coastline as well and would be appealing for kids. And the Roman museum at Vindolanda and a walk at the wall would be great.

I also don't have a lot of time to contribute, but have a look at this map and google the places on it - it's a possible "loop" from Manchester airport. https://goo.gl/maps/2TuoBKkRnwHg45EHA . They could also do a one-way drive, ending in someplace like Inverness, or do the same trip in reverse. However even with a fortnight some serious editing is going to be necessary; it's such a vast and varied (and marvelous) region of the world that neither two weeks nor two months would provide enough time to see even a fraction of the things available.