What evidence will I obtain that I have a double degree?

You will receive a degree parchment at your graduation ceremony which states the name of the degree that you have received and what universities have awarded it to you..  You should request transcripts from each university that you registered in to have an official record of your module results. For example see https://www.maynoothuniversity.ie/records/academic-transcripts-letters-and-forms for how to obtain your academic transcript from NUIM