Turns out there is a slight undocumented change in the latest PATCH SUPEE-6788 which caused problems for almost all of our Magento clients: the password reset link that got mailed to customers directed them to a page with no form to actually reset the password.

Turns out a change was made to customer.xml which causes this if you have copied over customer.xml to your own store views. customer_account_resetpassword was renamed to customer_account_changeforgotten and the block type was also changed.

The exact changed are outlined here:

customer_account_changeforgotten