I appreciate all the excellent advice I received on this forum for the trip from Australia in September this year to England/Scotland. It may not be helpful for anybody, but I thought I would recap it, and maybe it might be useful for somebody else going on a similar trip in the future.
15 nights to travel from Australia to London (4), York (2), Edinburgh (3), Lakes District (2), Bourton-on-the-Water (3) and Windsor (1) was busy, and we were tired at the end. I don't regret doing it, as we may never get back, but in hindsight, another couple of nights to take a break in the middle would have been advantageous.
Arriving in London at 6.30 am and needing to fill the whole day before a 4 pm hotel check-in is brutal. For the price of an extra night's accommodation, booking the night before is definitely an option.
The Heathrow Express booked in advance for 15 pounds each was easy, and I would recommend it.
We visited the tourist must-sees, including St Paul's Cathedral, the Tower of London, the Theatre, Buckingham Palace, and the British Museum. We also walked a lot around Piccadilly, where our hotel was. In hindsight, I am glad we visited the Tower of London, but I would rather have visited Westminster Abbey, which we missed.
The Underground was great - fast, cheap and safe.
We stayed at Thistle London Piccadilly in London. It is in a central location, and although it is very busy, we didn't have any issues with noise.
We did a London Walking Tour of St Paul's. The guide was great, so I would recommend this.
Going to the Theatre on night 2 was a mistake as we were still tired from the travel but a must do especially given we were staying so close.
We did the Buckingham Palace tour with the King's Gallery, Royal Muse, and Palace tour. It was not cheap, but it was worth seeing for the short time it was open.
Borough Market was a worth visiting and had a great atmosphere and in walking distance from the Tower of London.
We didn't have any fantastic food while in London other than an excellent curry at Borough Market, but everybody has different tastes, and we stuck with pub meals mainly on this part of our trip.
The LNER train to York was easy to find, on time and relatively clean. We had first-class tickets, but if on a budget economy, tickets would have been more than acceptable. We traveled on a Friday afternoon and the train was not full.