As we learn from the conversation of the two Weyland-Yutani Corporation officers in Hadley's Hope on LV-426, it takes 2 weeks to get and answer from Earth:
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/t7FZvTyf.jpg)
Assuming that the headquarters on Earth replies instantly, we can conclude that the distance is 1 light-week (as the communication signal would be travelling at the speed of light).
1 light-week is roughly 181 billion kilometers.
The fastest spacecraft ever built by humans as of now is the Parker Solar Probe which travels at 194 km per second. At this speed, it would take roughly 30 years to reach LV-426 from Earth.
Therefore, it would be pretty safe to conclude that the answer to the "how long" question is a few years the very least, depending on how fast that particular spacecraft of the future travels, and how long it takes the Earth headquarters to come up with the answer "don't ask" (that said, if they don't reply instantly, the distance will be shorter).
A period of a few years is also corroborated by the state in which the crew finds Hadley's Hope: the survivors would have been studying the aliens in the lab for several months the very least — after losing contact with Earth and apparently barricading themselves within the complex. It would then take some extra months/years for them to be killed.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/90k2JGKN.jpg)
The "three weeks" from the script would neither be possible (it would require the spaceship to travel at 100,000 km per second, which is 1/3 of the speed of light), nor would it make any sense to put the crew in deep frozen sleep for such a short period.
On the other hand, Rebecca "Newt" didn't age at all from the moment her parents found the trouble that killed the whole colony until she was found by the crew. Plot hole.