Okay so... ignoring the part where I might not be awake..
Assuming this is all on one ticket, you will check your bag at IAD and receive a bag tag which will tag your bags to EDI (assuming that your transfer in DUB is not overnight). I'm not sure, but it sounds like
garykung is right about needing to reclaim your bags for customs in DUB, but they should still have the EDI tag which would allow you to re-check them with EI. If you need to do this it should be really obvious - you will be routed past a baggage claim area, usually after passport control and before customs.
As EI is the marketing carrier of IAD-DUB, you are entitled to the EI allowance for IAD-EDI on the journey in your coupon class of service, and the UA counter in IAD will also grant you any waivers you might have based on your status with UA. Because it's a single bag tag, these waivers would carry through to EDI.