I was going to say that same thing. You can't be evicted like for something like that if you have a lease. Whoever told you that is dead wrong. Even if a landlord in the UK has given you an eviction notice, they actually have to go to court to get it enforced, and the rights that protect the renter are very strong.
As a social worker in the US and the UK, I've had to be aware of tenancy laws in both countries to help protect my clients. I actually feel that tenants rights are often more protected here than in the US. I've never had such strong protection regulations as those described in NYC.
All of my rental agreements in the USA (except for student housing) also worked as I've heard described here--a six month or one year contract that turned into month to month after it expired unless the landlord and I set up a new contract. I've never had one that automatically renewed to another year-long contract.