Can't you get a temporary one? I've just applied for one but if it takes too long they said I'll get a piece of paper instead of the plastic card. I'm applying for the E111 from Germany for the UK though.
It is "live" once your application ahs been acknowledged whether you are in possession of the actual card(s) or not. You just take a print out of their emergency number and the application reference with you.
You DO need it. I was in Spain recently when my nephew had to go to hospital. They insisted on a card, otherwise they'd have to charge for the treatment. He didn't have one but in the end they agreed to use my number instead and not charge us.
My card came back really quickly, around a week or so, but they must be fairly busy at this time of year.