Inbreeding vs Mutations
There are two major factors that will affect the development of diversity, given the same starting point: The mutation frequency, and the strength of selection.

Mutation frequency is unlikely to change significantly, so you're left with the strength of selection as the deciding factor for how much diversity arises and how rapidly. 

"Selection" means that less-fit individuals are removed from the population, so if the selection is strong enough, you're left with the race between evolution and extinction. With a small starting population and strong selection, the most likely scenario is extinction, but the right level of selection can lead to diversity.

An oversimplified view of selection is that a population will evolve toward some Platonic ideal. If long necks give more fitness than short necks, then the population will evolve toward having longer necks. The population changes, but isn't necessarily more diverse.

But there are several modes of evolution that drive a population toward increased diversity. In theory, you could imagine that the population does better with a 50/50 mix of long and short necks than either an all-short or an all-long necked population. Examples of drives toward population diversity include positive and negative frequency-dependent selection (also known as "balancing selection"), and overdominance.  

Are there any examples of bottlenecked populations where one of these drivers has led to increased diversity? Yes, there are several. The best example might be the Channel Island fox:


  The San Nicolas Island fox (Urocyon littoralis dickeyi) is genetically the most monomorphic sexually reproducing animal population yet reported and has no variation in hypervariable genetic markers.  ... foxes colonized the three northern Channel Islands (San Miguel, Santa Rosa, and Santa Cruz) ≈16,000 years ago and, subsequently, were transported by Native Americans to the three southern Channel Islands (San Nicolas, Santa Catalina, and San Clemente) 800 to 4,300 years ago ... We examine variation of five loci within the MHC of San Nicolas Island foxes and find remarkably high levels of variation. Further, we show by simulation that genetic monomorphism at neutral loci and high MHC variation could arise only through an extreme population bottleneck of &lt;10 individuals, ≈10–20 generations ago, accompanied by unprecedented selection coefficients of >0.5 on MHC loci. These results support the importance of balancing selection as a mechanism to maintain variation in natural populations and expose the difficulty of using neutral markers as surrogates for variation in fitness-related loci.


--High MHC diversity maintained by balancing selection in an otherwise genetically monomorphic mammal

The specific question "how long would it be before the population is free of problems related to a lack of genetic diversity?" is too broad to answer with any detail, but another point to keep in mind is that the problems of inbreeding are usually less to do with loss of diversity (although that can be one problem, as the San Nicolas foxes show) and more to do with deleterious recessives emerging as homozygotes; this can be softened by various factors, including chronic low-level inbreeding before the bottleneck occurs, which "purges" out the deleterious recessives and makes the population more resistant to inbreeding. One review: Do Plant Populations Purge Their Genetic Load? Effects of Population Size and Mating History on Inbreeding Depression