James Cameron's sub was a pressurized enviroment, a rigid container, he wasn't subjected to the pressure of that depth. A diver leaving the sub at that depth would no longer be in a rigid container, thus would be exposed to the full 101 atmospheres of pressure, and would start racking up decompression hours by the second. Max time with umbilical would depend on three things: how much decompression time they would be ok with, how large the gas supply is(since it is used at a rate 101 greater than at 1 atmosphere, note that the diving gasses required are rare, expensive, and in finite supply), and how good their thermal insulation is compared to the temperature of the water(for example even in a drysuit you wouldn't want to spend more than an hour in typical temperatures at those depths, because the temperature would be a uniform 4 degrees celcius unless it's near volcanic vents, that's the temperature at which water is heaviest).
PS: If the shelter is held at a lesser pressure than the ambient pressure at that depth, pretty much a requirement btw, then those month/year/decade long decompression times would apply to returning to the shelter as well as to going to the surface. Your divers would need to spend all that time in decompression chambers after a dive, if they wanted to return to the shelter.