Overall sounds like a good plan, well thought out. Just a few comments.
Paddington - not that central and a little run down (you will need to take the tube a lot), but perfectly fine - it does have some good and cheap places to eat. Hyde Park is lovely, and especially good with kids. Alternatively for a similar price range there are lots of cheap hotels around King's Cross Station. More central but much pricier are areas in the west end like Covent Garden, St James', Mayfair, and Belgravia.
Getting there from the airport - Heathrow Express is the most expensive options. Tube is substantially cheaper - you will have no problem with bags as the Piccadilly Line originates at Heathrow, so you'll be first on. Alternatively, take the Elizabeth Line to Paddington - it's almost as fast as the Heathrow Express and much cheaper. Just use your contactless card to pay.
Museums - I definitely recommend the Young V&A at Bethnal Green - it's great for kids. The Natural History Museum in South Kensington is also awesome for young ones.
Day 4 - Oxford Street is not what it once was - it looks really sad and run down. I would recommend Regent Street (which is nearby). For upmarket department stores, try Fortnum & Mason and Liberty. Harrods is in South Kensington, so nowhere near - it is however close to the Science Museum, Natural History Museum etc, so maybe club it together on that day. The area around South Kensington and Chelsea also has lots of nice smaller shops.
Day 8 has me confused - why stay near Heathrow? I would advise just staying in Central London and heading to the airport on the day of your flight. Windsor is an easy half day from London.
Other royal places not to be missed - Tower of London. Probably Kensington Palace as well (although not everyone loves it, including the people for live there!). On day 5, you could probably add a walk past St James' Palace (home to some minor royals, and the oldest working royal palace). Also have a walk through St James Park and across to Horse Guards, where you can see the changing of the King's Life Guard up close. These are troops of the Household Cavalry Regiment.
Highclere Castle is easy to get to by train from London, plus a short taxi trip to/from the nearest station. But I'm not sure how interesting it would be for kids. Also easy to get to by train from London and more impressive is Hatfield House, residence of the Marquess of Salisbury - check the opening dates as it's only open for a limited period in summer. There's also Woburn Abbey, residence of the Duke of Bedford (he has the largest art collection in private hands in the UK apart from the King).
Lots of great pubs in London, not a lot of space for patios - in summer it will mainly be drinkers standing on the footpath outside with a beer in hand. Country pubs tend to be the ones with beer gardens and outdoor dining.