Upvote:3
No, unfortunately this is not possible.
Even if you have residency in the UK, as a Cameroonian citizen you'll need a valid Schengen visa to visit Paris, and for that you need a passport.
In the longer term, if you've held ILR for more than a year you might be eligible to become a British citizen and get a UK passport, which would save a lot of visa troubles in the future.
Upvote:3
No, you cannot cross the Schengen external border without a valid travel document. The only exception to this is EU/EEA citizens carrying national ID cards.
For a Cameroon passport you'll also need a visa.