Drop Scotland.
Seems like you've booked your airfare. Have you done the same or assumed you could find something on the fly for your accommodations.
The currency exchange rate is at its best in years, London is one of the most popular cities to visit by all world citizens and your have chosen to visit during a busy summer month.
My first suggestions is to book your accommodations NOW!
As noted your trip is technically only 8 days because of arrival and departure days.
Given the ages of your children trying to see and do all that you've listed is far too much. The 4 year old will need nap time.
So:
Day 1: Arrive. Check into accmodation, freshen up. Depending on where you are staying ( what London neighborhood) will dictate what to do your first day besides dealing with jet lag.
Day 2: Parliament Square to see the iconic Elizabeth Tower a.k.a. Big Ben ( which is the bell), the Houses of Parliament. Then would the family more interested in Westminister Abby, riding the London Eye, taking a boat on the Rivers Thames, Horse Guard Museum, Churchill War rooms, seeing the Pelicans being fed in St James Park, Buckingham Palace, or photo ops near the Trafalgar Square lions?
ALL within walking distance of each other but you'll need to decipher which suit your interests more and time you will allocate to them. . Depending on your choices you'll need to have lunch and could always picnic in St James or Green Park. Take a long look at the satellite birds eye view on Google maps.
Do some planning and adjust your day accordingly.
Day 3: Morning train ride to Windsor Castle. Afternoon back in London. Maybe the Natural History Museum or London Transport Museum. Or paddle boats at Regents Park? Or seeing the buskers in Covent Garden?
Day 4: All day Harry Potter tour. Book your tickets NOW!
Day 5: early flight to Dublin. Check in at your accommodation. Tour the Guinness Museum. See St Stephens Square, explore Grafton Street. Dinner at a pub so you can enjoy your pint of Guinness.
Day 6: early train or bus ride ( 2 hours for either) to Belfast, visit the Titanic Museum, take a Black Cab tour of the Troubles then have them drop you at the train or bus station for your return to Ireland. This way you can say you visited 3 countries in 8 days.
Day 7 tour the Gaol and research to see if seeing the Book of Kells would be something the family would like.
Day 8. Shopping for gifts to bring home. Seeing whatever you missed.