You can find this number, which is usually a series of 10 digits, in several places:
On your American Express account statements:
This is the most reliable place to find it. It is clearly indicated on the paper or PDF statements you receive, often at the top of the page, next to the name of your establishment. It will be labelled ‘Numéro de Commerçant’, ‘Merchant Number’ or ‘Merchant ID’.
In your contractual documents:
The initial contract you signed with American Express to accept their cards contains this number.
On your online merchant portal:
If you have an account to manage your relationship with American Express online, your MID will be visible in your profile or account information.
By contacting the American Express merchant service:
If you can't find it, the simplest solution is to call the American Express UK merchant support service directly. Have your company details (legal name, address) to hand so they can identify you.
The phone number for UK merchant enquiries is 01273 675533.