Regrettably, I think it's extremely unlikely that your mother will be able to travel. The Passport Office take a missing passport very seriously and won't issue a replacement until they've carried out various checks (which is why the same day' service isn't available).
Unfortunately, your mother is a victim of the UK government's refusal to sign up to agreements which the rest of Europe are happy to enter into. Over 2 dozen countries are now signatories to the Schengen agreement, which permits travel without passport formalities between those countries. France is a Schengen member. If your mother was already in Germany, Greece, Malta or any one of many other countries, she could travel to France without any formalities. But the UK won't sign up, so we all have to go through passport control (Under completely separate rules, anyone with a national identity card, from any EU state, can freely enter any other EU state but our government still can't decide whether it should introduce national ID cards or not).
No form of visa will help because a visa isn't valid without a passport.
The only slim chance is to phone the Passport Adviceline as soon as possible, to see what they suggest. It's available 24 hours a day, so do it NOW:
0870 521 0410
Chris