Upvote:1
Assuming it's all one airport (eg NYC-LHR-AMS or similar), if you are allowed onto the flight in New York, you'll be fine
As long as it's all within one airport (so no Stansted-Gatwick changes or similar), and all your bags are checked through (so no need to clear customs + immigration to get bags to re-check), you won't see UK immigration on a transit.
That isn't to say there are no checks!
The UK, in common with many countries, requires visas from many countries even for a direct airside transit in the UK. See gov.uk for more details. As part of that, airlines have to check eligibility for transit without visa, and have to forward passenger details to the UK in advance, so the UK can deny boarding for people not allowed to enter.
If you have to enter the UK to change airports or re-check bags, you'll have to pass UK immigration, who'll see your previous issues and most likely deny you. They'll then most likely put you back on a plane to the USA, maybe even at your expense. From your perspective, this would be bad! If you can do it all airside, it depends on "how naughty you were". Moderately naughty and you shouldn't be on a black list, so should be fine to transit and depart.Too naughty in their eyes and the airline will be told to deny you boarding.
Safest would be to change to a flight from New York to Amsterdam that's either direct, or changes in another Schengen country. You do still risk intra-EU information sharing on previous entry denials, but assuming you have a valid and non-infringing reason to be in Amsterdam, you should be fine to enter that way