I'm a green belt. Without wishing to be pedantic, asking how long it would take you to get a brown belt is like asking 'how long is a piece of string?' It depends on how often you train, how much prior experience you have, how much natural talent and athleticism you have, how good your instructor is, how good the other students are, how well you apply yourself, etc, etc.
There is nothing worse than having a belt you don't really deserve so don't rush it, it's only a belt after all. There is so much to learn with judo (as with every martial art) so whatever colour your belt is there will always be more you can learn. Having said that I met a guy recently who got a back belt in 3 years so it can be done.