The latest paper that I'm aware of on this topic deals with a very detailed calculation of calculating the tipping time of a quantum rod. In this calculation, the author showed that the classical problem can be recovered when the Planck constant goes to zero, and draws the conclusion that:
.. the tipping of the quantum rod can be understood as having been triggered by the uncertainty in angular momentum engendered by localization of the initial state...
The article is a bit difficult to follow, and I didn't get any direct value of the tipping time.
The more interesting papers that I've found earlier on the same topic are much more illuminating than this one. A paper by Don Easton presents a caution for people who tries to apply QM as the basis of the tipping time. His calculation of the tipping time, using QM, gives a humongous number: 0.6 million years. He examined why some posted solutions actually gave a balancing time of the order of 3 seconds, and why those treatment may be faulty.
Another paper that cautioned the use of the HUP in calculating the tipping time is a paper by Shegelski et al. Here, they caution that one can't just use the HUP alone, and they also compared this to the faulty application of the WKB approximation to this problem.
Fascinating! Certainly something that I read in bed before going to sleep! :)
 O. Parrikar, Eur. J. Phys. v.31, p.317 (2010). You can also get a free copy of the paper within the first 30 days of online publication at this link.
 D. Easton, Eur. J. Phys. v.28, p.1097 (2007).
 M.r.A. Shegelski et al., Am. J. Phys. v.73, p.686 (2005).