The Portuguese system works a bit differently...
All visa applications lodged in the UK need to made online at their Application Portal.
As part of the procedure you will have the opportunity to schedule your appointment request and the site will tell you which locations and dates are available. Edinburgh, being an honorary consulate, does not have the credentials to issue visas. The choices in England are London and Manchester.
You should study their guidance (linked from the portal site) and be sure to take your original documents plus a copy of each (also your photographs).
They have an background information page here.
In researching this answer I spoke with the Vice Consul at The Consulate-General of Portugal in London. She emphasized in the most emphatic terms that you should not purchase travel tickets until AFTER your interview.
In the event you have further questions you can contact the Serviço de Estrangeiros e Fronteiras (SEF) at firstname.lastname@example.org