A quick search shows that it's generally listed as dovecote, but mostly where the other spelling is concerned it's "Dovecote, also Dovecot." I've always* used 'cote,' but suspect that 'cot' is the spelling after contracting terminal American Spelling Syndrome.

Just like poor aluminium.

* When I say 'always,' I mean - just how often would one actually need to speak about dovecotes? O_o