After the bill is transmitted to the Governor, he has 10 days--Sundays do not count. http://www.capitol.tn.gov/about/billtolaw.html
Signed by Speakers
After the Speakers sign the enrolled copy, it is automatically transmitted to the Governor for his action.
Signed by Governor
The Governor may sign the bill; veto it; or allow it to become law without his signature. The Governor is allowed ten days (Sundays excepted) after a bill is presented to him to approve or veto the bill; if he takes no action within that period, the bill becomes a law without his signature. The Governor also has constitutional authority to reduce or disapprove any sum of money appropriated in any bill while approving other portions of such bill.