Sorry to hear about the rain, but hopefully you can salvage the latter part of your trip. Nice would make a great spot to spend a few days, and there are some great photo op's in that area. However, I'm not sure how easy it will be to find hotels on short notice at this time of year. For a 30th anniversary, I can't imagine a nicer place than http://www.hotel-nice-suisse.com/uk/boutique-hotel-nice-mer-site-officiel.php (be sure to ask for a room with a view of the beach).
If you're flying, the airport is not far outside of Nice so transportation is easy. For travel back to Milan for your flight home, there's a direct train departing Nice Ville at 08:09, arriving Milano Centrale at 12:50 (that assumes you're going to spend the night prior to your flight in Milan). That's an EC train so you'll need reservations. I'd probably try booking on Capitaine Train as soon as you have your plans locked-in.
Don't be too concerned about not speaking French. As the others have mentioned, do your best with the "polite words" and that should work fine. Most people working in the tourist industry will be able to function in English to some extent.
For sightseeing in that area, you probably won't have the appropriate RS guidebook along. If you're travelling with a Laptop, Smartphone or other device that can handle PDF files, you could try downloading the appropriate Lonely Planet chapter at http://shop.lonelyplanet.com/france/france-travel-guide-11/france-the-french-riviera-monaco-chapter-11/ ($4.95, 11th Edition, March 2015).
Good luck and congratulations on your anniversary!