Upvote:0
Your mother should explain her itinerary and premise for the visit. As I understand it, the trip would be
China -> UK -> France -> UK -> China.
For the Schengen visa, it greatly helps that she has an UK visa and relatives in the UK, since it would be rather implausible that she wants to overstay in France. If she had any such thing in mind, that would be in the UK. You should definitely submit a copy of the UK visa.
An invitation by an UK resident for going to France would be kind of contradictory, unless you are paying her travel expenses, but the fact that her daughter travels with her should be mentioned to explain the itinerary.
Upvote:1
It can be helpful to provide an informal letter from you and your wife about the plans; this helps establish credence for her stated travel plans, which is always good in a visa application. The point of such a letter would be to evidence that the people she claims to be visiting do indeed exist and provide contact details in the (unlikely) case the consulate decides to verify it. Do not attempt to promise things in the letter that are not yours to promise; in particular, explicitly promising that the mother-in-law is going to leave the Schengen area will make the letter less helpful.
Unless you and your wife are funding the mother-in-law's visit, that should be all you need. If there's already an UK visa in her passport, the French consulate ought to assume that the British have already checked whatever they want to check about her host having legal status in the UK.
If you are funding her trip, then the consulate will want to see some documentation for your financials -- bank statements, etc. -- to make it look plausible that you can afford (and would choose) to spend money on her trip.