My wife's sputum test took either 6 or 8 weeks (can't remember now) because she was pregnant and they wouldn't do a normal x ray where results are told that day.
As for working, like has been said, as soon as you get a national insurance number, you can work right away on a spouse visa.... My wife started work last month, been here 11 months now, but the national insurance number we sorted out within a month of her arriving here.
Edit, dug up my old post regarding the sputum test, was a longer wait than I remembered!
Going for the Sputum test, it is done over the course of three days.. Couldn't do it this week so she is going Mon, Tue and Weds next week.... Then we have to wait 8-9 weeks for the result....