I lived in London for 3.5 years and traveled often to the US before the pandemic started. Here is my advice if you cannot switch your plane tickets…..do NOT take a black cab from Heathrow to Kings Cross Station!!! You pay a FORTUNE!!! All black cabs charge a metered fare vs a fixed price. I only used them when I had to quickly navigate around London. We always booked our airport transfers with Addison Lee, and will be doing so again when we travel back this Summer for a visit. Never had an issue, and the drivers were always professional and knowledgeable. You should pay around £50 for the journey.
The Piccadilly line is the cheapest option however it takes over an hour with all the stops, gets very hot in the Summer, and can be crowded!!! Remember that you have all this bulky luggage with you too, and people often just shove in the Tube to get off at their stop. We did it once and never again!
The information desk at King’s Cross is fantastic for booking tickets since they know the best group deals/ getting a national rail card. Also this website is the best for understanding everything to know about rail travel in England and to Europe. https://www.seat61.com/
Keep in mind that the train ride is 5.5 hours from King’s Cross. It is already a long overnight flight to reach England, and then to navigate to Kings Cross and make a train to Scotland makes it a two day marathon. Why not stay one night in London and take the first train the next day? London is a clean amazing city with so much to do, see, and eat! Even if you just book a hotel by King’s Cross and didn’t venture far, you would still enjoy your journey more in the end.
Hop on the Northern line to Hampstead, eat crepes at the local food truck, book a table at the Wells Tavern for the BEST sticky toffee pudding in all of UK, and have a flat white and cinnamon social at Ole & Steen. Or grab picnic items at Karma Bakery, and join all the English locals relaxing in Hampstead Heath. Enjoy your time with your mother and sister!