What is the train schedule Basel to Florence, Italy?

Go to bahn.de for schedules of all European trains. That way you can find the schedule you like.

The query page for the Bahn is actually here. Franks link will get you here eventually, but you might as well start with the query page. The Bahn shows a trip of a little less than 7 hours with changes in Zürich and Milan. There are trains about every two hours.

or via Bern (no change), Brig (no change), change in Milano. Less scenic because of the time spent in the Basis Tunnel but the trip is under six and a half hours.

The Swiss rail site, sbb.ch or rail.ch for English, will show you more options and the prices between Basel and Milan, as well as offer specially priced tickets for advance purchase.
The direct Basel to Milan route via the Basistunnel is indeed faster by 45 minutes, but the departure time from Basel, 12:31, gets you in to Florence pretty late. The price for this train is 109 CHF full fare, 27 CHf for the Special Offer Price. The Gotthard Pass route has a change in Zurich and takes 45 minutes longer getting to Milan, but the route is more scenic and has more convenient departure times---8:07 or 10:07 am. You arrive in Florence about 6.75 hours later, so even the later departure woud get you to Florence before 17:00 (5 pm). Prices on this route are 116 CHF full fare, 57 CHF for the Special Offer. You can find prices and special fares for the Milan to Florence portion on Trenitalia. Maybe as low as 9 euros if you get a Super Economy fare.

Basistunnel route also at 6:31 with arrival at Firenze SMN at 12:55.

True. But there is no way I am going to get up in the middle of the night and skip breakfast to catch a train that early! So I do not even look at the schedules that far back. Maybe that time is OK with Nancy so it is good you mentioned it. It would give them more time in Florence.