How much does it cost to phone 001 international dialling code from the UK?

That depends on the next three digits after 00 1.

00 is the international dialing prefix for outbound international calls from the UK. Telephone country code +1 is North America, which includes the USA and Canada, but also includes 18 miscellaneous Caribbean/nearby island nations and territories. The rate for calls from the UK depends significantly on which North American country you are calling. In general, the rates will be lowest for the USA and Canada, and may be dramatically higher for the Caribbean.

Of course, your costs will also depend significantly on your long distance provider and service plan.