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Tax haven companies/trusts can open bank accounts anywhere?

Technically yes, practically it is almost impossible nowadays to open a bank account for a company registered in a tax haven.

Try to open a high street bank account in London for a UK Ltd company with non resident director. All banks will reject you.

Only when the structure makes sense, enough money is involved and with some industry experts one can make a chance, but I doubt in London or New York. Singapore and Hong Kong banks are much more open to offshore corporations banking
 
It's all about money. The amount of money you bring, where the money comes from, where the money is going, and how much money the bank can make. That is then set in relation to the risk profile of the applicant.

There are Cayman companies/entities banking in London and New York. It doesn't mean some random person has any realistic chance of showing up with their Cayman company. Those companies that have accounts there have them because they are part of VC or PE structures, investment vehicles for very wealthy clients, and so on and so forth.