With all the talk of Roy Moore and Doug Jones, I’m surprised this is as difficult to discover as it is, but I haven’t been able to find it: once the victor, whoever that is, is determined in the 2017 Alabama special Senate election, when does that person actually become Senator, i.e. able to vote in the Senate (and, by the same token, when does Luther Strange lose his appointment and become unable to vote in the Senate)?
When will the winner take office?
Likely in early January. First, Alabama has to certify the results. Its 67 counties have until December 22 to report their results to the State Canvassing Board, which is made up of three Republicans: Gov. Kay Ivey, Secretary of State John Merrill and Attorney General Steve Marshall.
The earliest the state could certify a winner would be December 26, but the secretary of state's office does not expect the canvassing board to be able to do so until January 3, and it could be even later, delayed by slow canvassing by the counties and jam-packed holiday schedules between Christmas and New Year's Day. Once it does, that certification would be sent to Washington immediately. Once the winner is certified, the Senate would likely in swear that person quickly.
A delay of two or three weeks between Election Day and being seated in office is normal. Sen. Scott Brown, a Massachusetts Republican, waited 16 days from his election to his swearing-in. Sen. Cory Booker, a New Jersey Democrat, faced a 15-day wait when he was elected in 2013. And Sen. Ed Markey, a Massachusetts Democrat, waited 21 days between being elected to replace John Kerry and taking office in 2013.
when does Luther Strange lose his appointment and become unable to vote in the Senate
When the Senate accepts the new Senator. So if the Senate does not recess because there is a government shutdown, Strange might still vote through the rest of this year.
In an extreme case, Al Franken was not seated for six months in 2009. Note that that was in addition to the normal two months between the election and taking office. Because this is a special election, the term starts as soon as possible. Potentially as quickly as two weeks, as shown in the @Machavity answer.
As a practical matter, if the Senate is in recess when the Senator first becomes available, the winner will have to wait until the Senate can accept the state's certification. But while Strange might technically be Senator during that period, he wouldn't be able to vote, as the Senate would be in recess.