By train this is a real pain of a journey, 9-11 hours with anywhere from 3 to 5 changes. Realistically the only way to do it would be to fly to Geneva from Nice and transfer by road to Chamonix. EasyJet and the Swiss LCC Baboo fly this route. There are various companies that operate transfer service between Geneva and Chamonix. If you use the search feature above the posts and search on "Geneva transfer" without the quotes you will find a wealth of information on the various options.
I know you said you were not renting a car but that would be the easiest way to get to Chamonix, either driving from Nice, about 5 hours, or flying to Geneva and driving from there, about 1.5 hours.
I agree with Irishrover
Thank you so much!!!
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