Birthday and names can come from ticket data.

Even easier... Prior to this, he had asked audience members to write down any questions they had on cards he handed out and seal them in envelopes, write their seat number on the envelope and put the envelopes in that bowl on the stage.

I'm presuming the cards asked for more data than just the question. They probably also asked for birthdates, and some additional information (like "Anything interesting that happened to you recently"). He just needed someone to go up there pretending to drop a card in that bowl, and to instead pick up a few cards and feed him the information through an earpiece concealed in the bandages.

Anyway the "Oracle act" is a classic mentalist trick with many variations. Google "Q and A routine". Derren Brown performs it with much more flair than most, but he's not doing something completely unknown here.

I'll note that the act is from a stage show that is filmed in its hour-long entirety, so its unlikely that these are just snippets of longer conversations. Also, he specifies beforehand that he doesn't use stooges or have any prior contact with his audience. I doubt he's lying about this.