The UK doesn’t do exit checks or stamp passports on exit for anyone. Your girlfriend can add her ID card data as advanced passenger information on Ryanair or any other airline, and that is sufficient to leave the UK to go to the EU/EEA.
The only risk is that she cannot get back into the UK without a passport. Assuming her British passport is approved, you can receive delivery of it and either bring it to her or post it to her.