You can convert Amex points directly into Nectar points at a 1:1 ratio (1 Amex point converts to 1 Nectar point).
But, there's a trick that means you can actually get 1.33 Nectar points for every Amex point you have, though you have to convert Amex points to Avios points, and then to Nectar points.
Here's how to do it:
Swap your Amex points for Avios via the American Express website.
Link your Nectar and British Airways Executive Club accounts, which can be done on the Nectar or British Airways Executive Club websites. Both are free to join if you're not already a member.
Finally, convert your Avios to Nectar points via the British Airways website. You can only covert in multiples of 300 Avios (300 Avios = 400 Nectar points).
You can then spend the Nectar points at any Nectar partner, including Sainsbury's, eBay, and Argos.