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Mine appeared 24 hours after receiving the jab.
I think you are also depending on the Spanish government allowing visitors from the UK which I don't think they currently do. At the moment its only EU and Shengen allowed in I think.
I can't find the details for Spain but I understand several EU countries don't require the 2nd jab for a person to be considered vaccinated. The require 1st jab plus a certain number of elapsed days. Hopefully in your case, Spain is one of them.
Anyone know if the record of your vaccination appears immediately on your NHS app? We're due to see my wife's family in Spain in August and her second one is one day before we travel and is on a Sunday, so time couldn't be tighter. I only downloaded the app today and my second one was last month so both of mine show up obviously; I feel like it should be pretty swift as the GP seemed to know straight away for me.