In Madrid, everything needs an appointment when it comes to government services. Check the Ayuntamiento Madrid site for the empadronimiento and the contact numbers to arrange for the appointment. They also have details of the documents you need - EU Passport (if you have one), NIE, etc.
The certificado de registro de cuidadano UE - I don?t understand why you need this if you are an EU Citizen? Do you have an EU Passport? If you?re not an EU Citizen, but an EU Resident due to marriage with an EU Citizen or family links, that?s a different story, and you?ll need to check with the Spanish Consulate in your country on what you need. Actually this would be important to get your NIE, I would presume!
For example, I?m a non EU with residency through my Spanish husband, but I still need a visa (a spouse visa) to stay in the UK. If you are a UK resident but non-EU/UK, then you should double check if you need a visa. If you?re a EU resident, non-UK, then of course you should be able to enter Madrid no worries. I would presume you would have some document proving that you are a resident of EU and maybe they?ll accept that? You should check with the consulate.
Of course if you have a Spanish friend or you speak Spanish, it would help, because they absolutely DO NOT SPEAK ENGLISH in the foreign registry offices. So the spanish speaker can help a lot to get you the appointments, etc. If you don?t have any spanish support, perhaps getting a gestora like Campana mentioned will be a really good idea (saves you a lot of time, sweat and tears… hhaha as we all know the bureaucracy here is a nightmare).
Good luck!! 😊