First off, you’ll be fine. Heathrow handles tons of solo travelers every day.
1. Immigration
Since you’re 16, you won’t use the e-gates. When you arrive, just follow the big “Arrivals” and “UK Border” signs. There will be two clearly marked areas: e-gates and “All Other Passports” or desks with officers. Staff are everywhere directing people, so it’s very obvious where to go.
They’ll likely ask simple questions:
Why are you visiting?
How long are you staying?
Where are you staying?
Who are you staying with?
Have:
Passport
Address of where you’re staying
Return ticket details
Parent consent letter (good that you have this)
Your sister/relative’s contact info
Just answer calmly and clearly. It’s routine.
2. Getting to King’s Cross
From Heathrow, follow signs for the Underground (Tube). It’s very well signed. You can absolutely tap in and out using a contactless credit card or Apple Pay. No need to buy a paper ticket.
3. Which train?
The Piccadilly line goes directly from Heathrow to King’s Cross. No transfers. Around 50–60 minutes.
At 11am on a Thursday, it won’t be rush hour chaos. It may be busy, but not “can’t move” busy. You’ll be fine with a suitcase, just stand near the doors.
Other option:
Heathrow Express to Paddington (faster but expensive)
Then Tube to King’s Cross (adds a transfer)
Honestly, for simplicity at 16 and solo, I’d just take the Piccadilly line direct. Cheap, easy, no transfers.
You’ve clearly done your homework. You’ll be okay. London transport is very straightforward once you’re there.