As stated above, if it's the "Flexi" fare, then lounge access is included (and you should have received a separate email at the same time as the e-ticket was delivered, the separate mail containing a lounge pass voucher to be printed off. That you didn't receive this makes me think you're not on a Flexi fare)

It used to be that a BA Gold or Silver card was enough to get you into the Gold Circle lounges in Dublin and Heathrow for ANY flight between Dublin and London, even when booked through Aer Lingus and having no onward connections with BA. All of the Aer Lingus flights carry a BA codeshare number; even if you're not travelling on the BA code (which you can only obtain if you are connecting to/from another BA service at Heathrow) BA apparently were happy to pay for their Golds and Silvers travelling on the route in any capacity (i.e. even when it was a pure Aer Lingus ticket). For as long as I've had a BA Gold card I've never failed to get into the GC Lounges in either Dublin or Heathrow, but it's not a route I travel much and I've not travelled on this route since BA took over the bmi services on the route. So perhaps the access rules have changed. But it's certainly worth a try at the Gold Circle lounge in Dublin - your BA card and your EI boarding pass for a DUB-LHR may still get you in!