===> "What is the minimum amount of time a manager can be working in order to have an affect on the club (positive or negative)?"
Negative - not too long IMO and IME.
Positive - potentially many months or longer, depending on all of the factors 25 said. Especially once a place develops a bad local rep for being dangerous and/or putting pigs and fuglies on stage. I've seen it happen twice with two small local clubs and in each case it took a year or more.
Each time I witnessed it, they first made the club safer by implementing tighter security at the door and, when necessary, on the floor. This is the quicker fix though. Once you make the place safe for old men to spend their money in, then it is always and forever about the girls.
The problem then becomes one of the chicken and the egg. Hot girls get impatient quickly when they struggle to make money in a club, but it's hard to bring good money in when you don't have them for a while. This is where a manager's real skill, or lack thereof, shines.
In each case I witnessed, the managers slowly built up the roster by making their clubs easier to work in than anywhere else. Easier to come and go, no drama or micromanagement, etc. Believe it or not, a certain % of girls will actually sacrifice a certain level of earnings for a place they feel appreciated and comfortable in. One of these clubs also beefed up their roster by serving as the place of last resort for hot girls who got fired from the other local clubs.
All of this of course came with a little more behind the scenes drama than some managers will tolerate. But when you're trying to overcome a bad rep and are competing against other clubs with strong talent and perhaps better facilities, you have to make some tradeoffs.
Then it is just a function of time. More guys come in and keep coming back because they like the scene. Customers follow their "last resort" dancers. Etc., etc.