I figure the easiest way is doing this by train.
Take a TGV service from Nice to Paris, this will take just over 5 hours but no need to go to and from the aiport, no advance check-in, etc.
From Paris take the Thalys service to Amsterdam (check whether this is possible with a Eurail pass). From Amsterdam take a cheap flight to London on easyJet or another carrier.
The Eurail pass will costs around 270 USD, there's no way you will find an airtickt NCE
for this amount of money.
Ryanair only works when you have a car or don't mind ending up travveling long distances to and fromt the cities mentioned above.
You can try easyJet for the NCE
-PAR segment and book a single ticket for the Thalys Paris-Amsterdam (no Eurail pass needed) then fly AMS
-LON on easyJet.
(pick the ticket up in Paris)