This is my understanding.
Neither has priority. Both should apply unless one or both renounces.
You could ask the solicitor to renounce their entitlement to extract the Grant. Your father did make the will however and he did appoint the solicitor. The solicitor does not have to renounce and it is unlikely a Court would remove them.
If there is no charging clause in the will ( unlikely) then the solicitor could not charge.
If the other executor renounces, your mother could then extract the Grant with you effectively doing everything.
Your mother is not the Testator and it is not within her power to ask you to administer the estate. However, if she renounces and the other executor renounces, then if you are mentioned in the will as a residuary legatee, you could apply for Grant of Administration Will annexed.
mf