The CR95HF chip is listed on a.o. Digikey and Mouser. Probably without PCB, though - a quick search on Taobao showed CR95HF based boards; they must be available on aliexpress and likely e-bay and co. as well.
Most BT modules communicate over
jwarnes:
The first Arduino unit I thought I might try would be a nano, and I have no idea which Bluetooth unit to use. Does the Bluetooth unit and Arduino unit need to talk to each in SPI?
Serial, usually.
And what software will I use on the phone or smartwatch?
Probably software you wrote yourself...