How do I upgrade a HP Envy 5660 printer from 32 bits to 64 bits?

Printers themselves are not any number of bits.

The HP ENVY 5660 e-All-in-One Printer is Apple AirPrint-enabled. No drivers are required. Do not install drivers or other software supplied by the printer manufacturer.

If something is complaining the driver is not 64-bit compliant, then remove that printer from Printers & Scanners and add it again. You must specifically choose AirPrint. Refer to Add a printer on Mac.

You don't. HP must provide 64-bit drivers for that model. They show new firmware released in Oct 2019 and a new driver in in Dec 2019:

https://support.hp.com/us-en/drivers/selfservice/HP-ENVY-5500-e-All-in-One-Printer-series/5304881/model/5304882

There is not specific mention if they are 64-bit but that it a question for HP.

EDIT: a but more clicking on that HP page shows the new software is compatible with macOS 10.15 Catalina, so it is 64-bit.

You will have to visit the HP support site to find and download updated drivers.

HP Software and Driver Downloads for HP Printers - HP