Switched computers, and now eM Client is telling me I have reached the maximum numbers of activated installations and won't recognise my key

I recently switched computers, as my previous laptop completely stopped working.

Now eM Client is telling me I have reached the maximum numbers of activated installations and won’t allow me to enter my license key without updating the license to purchase more, or deactivating other installations. 

Surely the license is linked to my email address, and not the previous laptop? I don’t have access to my old laptop, and need to activate the pro version of eM Client on this laptop ASAP. 

Go to emClient licensing server. There you can deactivate all unused installations,

The licence is registered to an email address but it is bound to a specific computer hardware id. When moving the application from one computer to another, or before changing the motherboard in the computer, you should first deactivate the license.

If you did not do that, then as Pefunk commented, you can go to the licensing server and deactivate it yourself.

This is only for Pro Licenses. For a Free License you can either write to eM Client and ask them to deactivate it for you, or you can register a new license using another email Address.