Ciao Strambi and welcome back! Wow, 11 years ago, are you coming back for your anniversary?
If you have 10-15 days between April and May, I'd suggest May. Less chance of showers, getting nicer temperatures in general. April is still a very good choice, but later just makes it more likely to be warmer. Although you can't count on anything, since this past June was very rainy!!
So no Florence.... then how about setting up a base for 7 days in one place, then moving on to another for the remaining time?
I suppose you will rent a car to make it easier to move around? In that case, you can stay in the area to the southwest of Florence that is still considered Chianti and use that as a base to go west to Pisa, Lucca, San Gimignano, San Miniato, Certaldo and east to hilltops towns in Chianti such as San Donato, Castellina, Radda, Panzano and Greve/Montefioralle.
Then move on to your second base in southern Tuscany and from there visit that area: Siena, the Valdorcia towns of Pienza, Montepulciano, San Quirico, Montalcino, Buonconvento, if you want to visit thermal hot springs you can go to Bagno Vignoni and San Filippo. You can also head east to Arezzo and Cortona, or even west to the coast to the Maremma area... AS WELL as UMBRIA.
You can visit Peruggia, Assisi and Lago Trasimeno from Tuscany as your base. Depending how much you want to see in Umbria, you could take off stuff mentioned above and add to Umbria. If that is the case, then you could cut this week shorter and move on south for a few days, rather than do it all from Tuscany.
15 days will not be enough!!
How does that sound for now? Keep posting any questions as you move forth and need any assistance!
And thank you for thinking of us and coming back to plan your trip back to Tuscany!