Would I need to show them my birth certificate?
There is a page that discusses whether you need a birth certificate to apply for a passport. It says that one condition that relieves you of this requirement is:
vous êtes né(e) à l’étranger et votre acte de naissance étranger a été transcrit dans les registres de l’état civil consulaire français.
you were born abroad and your foreign birth certificate was transcribed in the French consular civil register.
I presume that this second clause is what happened when your "birth was registered at the French Consulate in Liverpool." If so, you don't need to show your birth certificate to apply for a passport. I suppose that you might in this case require a document recording or attesting to the transcription of your birth certificate, but I couldn't find anything saying that explicitly, so maybe you do not.
And would I also need to show a scanned copy of my Mother's birth certificate, or must I provide them with the actual one?
It's not generally necessary to show your parents' birth certificates with a passport application, but if you are under 18 you'll need to show your mother's passport or identity card and livret de famille, the requirements for a first-time passport application for a minor being somewhat different from those for an adult.
I assume here that the registration of your birth with the French consulate serves as your proof of French nationality. If it does not for some reason, you will need to have your French nationality certified as a separate step before you can apply for a passport. Doing that will of course require proving your mother's nationality, in which case her birth certificate may be of use.