How to glue mug pieces
Most 2 part epoxies are food safe once fully cured. You can see these used in kitchen applications like counter tops and bread boards. There are some that advertise as being food safe grade. 


Masterbond.com


Reading the product labels both on the product itself and online will help. Some will even reference FDA compliance when it comes to food safety. If you find something that does not quote it, err on the side of caution and consider a different product. 

Many epoxies, once cured, can handle high temperatures. So it should be fine to use to glue your mug pieces. Ratings of different products will vary so be sure to make an informed purchase e.g: 


3M™ Scotch-Weld™ Epoxy Adhesive EC-2216: Rated to -423°F/-252°C
Masterbond EP21TDCHT: Rated to -100°F/-73°C to +350°F/176°C
LePage® Epoxy Gel: Rated to -9°F/-23°C to 120°F/49°C


Note: the latter is not rated for something like boiling water. 

They can take from 24 hours to a couple of days (usually for larger applications). I recently used epoxy to fix my hand made ceramic renaissance mug handle which broke in 4 places.