BA OLCI will open 24 hours before departure of the first flight, the EI one, at that time you can also check in to the BA leg.

It should though not 100% offer an ability to check in via BA for the EI flight or tell you to use EI's system. As you manually added a seat this may mess things up or make it work better, I don't know. I do know for EI tickets which a seat edit is performed by an agent 72 hours or nearer to departure it may block OLCI.

If it doesn't work the self service machines should be happy to check you in. A human may be required if your destination has specific visa requirements