100,000 years (or 100,000 generations), give or take, assuming ideal conditions
So, let's get the math out of the way first; the first animals of the human genus appeared on earth around 2.5 million years ago. Assuming that we are relatively new on the scene, and we are estimated to have first appeared around 100,000 years ago, that would mean that it has taken us around 2.4 million years to evolve from our first ancestors. If we make the math easy on ourselves and assume that a human 'generation' is 24 years, that means that we evolved in 100,000 generations. That assumes of course that the early homonids were almost cat-like in their levels of intelligence and interaction with the environment, which is a large assumption to make.
Cats are capable of gestation within 6 months, but (again to make the math easy) let's assume that the average feline generation is 1 year - if so, and if it takes the same number of generations to achieve humanoid intelligence and traits, we're looking at around 100k years for it to happen.
There are some massive gotchas in that estimate though. First, cats can live for 10 to 20x past their first batch of the next generation, and they remain fertile for a lot of that. There would be a massive cross-fertilisation of 'generations' as a result meaning that cats would be unlikely to have generational progression on the same rate as humans.
Additionally, humans have very few children by comparison, live for only around 3x their age at their first generational production and are fertile for a lot less than that. This is important because it means that there is not as much cross-fertilisation across the generations but even more importantly, less competition for resources between generations. Cats are going to live for some time, constantly cranking out offspring, meaning that there is a high chance of a population blowout that could wipe out the species through stripping of all resources if a given generation proves sufficiently prolific.
Also, cats are already well adapted to their function as hunters, there are few threats that would evolutionarily reward the high expenditure of energy in neural enhancement like what humans were adapting to. Cats are already 'intelligent enough' for what they do and it would take a very specific set of circumstances to make intelligence more important than their existing traits in hunting.
What is far more likely is that cats will run out of food and begin to compete with each other if their numbers increase in an uncontested environment. Intelligence may be a desireable trait that is rewarded with survival in such a case, but it is rare enough that I doubt it. That said, cats would have a far better chance of developing intelligence if they had fewer offspring as it would mean that the previous generation can't overwhelm the next one with numbers that make intelligence irrelevant as a survival trait.