There isn’t really a typical response time. Unlike Ryanair, who’ll respond and quite directly tell you to take a hike, Aer Lingus tend to just ignore complaints or queries that they don’t want to respond to and stonewall. If you’ve not heard back by now you’re unlikely to hear from them. You can try sending a snail mail letter to the CEO at Shamrock House, but that’s not really guaranteed to get a response either. At this stage, it’s reasonable to assume they’re not going to do anything for you unless they’re forced to.

While it's probably unwise to travel with a baby with no documentation whatsoever, it does appear from EI's site that they have no requirement for any, so you appear to be technically in the right. An airline is likely well within its rights to ask for documentation if anything arouses their suspicions and deny boarding if they're not satisfied, but given they don't state upfront that documents may be required, I'd argue they should have been more willing to work with you on this (eg, get someone to send a copy of the birth cert, then move you to the next flight for free).

They’ll usually defend themselves in a Small Claims action, but that’s not to say they’ll win, so not a bad course of action if you feel you have a case.

You could also try contacting Pricewatch at the Irish Times. While Aer Lingus hasn't cared about their appalling phone and email customer service for many years, they seem to be quite sensitive to how the public perceives their customer service. I guess it's hard to claim the "You're very welcome" banner when there's a steady procession of horror stories in the paper about your service.. Pricewatch has generally been very good at leveraging that to get them to fold quickly with a vague "in this very rare case we've not lived up to our usual standards" kind of response.