The rule stated in the Directory for the Application of Principles and Norms on Ecumenism is:
133. The reading of Scripture during a Eucharistic celebration in the Catholic Church is to be done by members of that Church. On exceptional occasions and for a just cause, the Bishop of the diocese may permit a member of another Church or ecclesial Community to take on the task of reader.
This is what the Church says with regards to this matter. It’s not uncommon during certain ecumenical times of the year, another Christian may be permitted by the bishop to do a reading. Personally, I dread this because usually the person gets up to the ambo and starts off telling a story about how happy they are to be there, etc etc…
Anyway, you don’t need to worry about no one being able to serve as lector in your family. Parishes have people whom they can call on to provide this service.
Funerals and weddings are the times when non-Catholics may be allowed to read the First and Second Readings and the Prayers of the Faithful. While it belongs to the Bishop to give permission, he may have given the priests of his diocese his guidelines and let them decide each case.
I have to admit that I’ve never known a family to be asked the religion of the persons they had chosen to do the readings at weddings and funerals in this area.