As indicated above, a lot depends on the flow rate the first stage is capable of maintaining.
Reg manufacturers figure max flow rate for a first stage by essentially removing one of the LP ports and letting things rip. A Scubapro balanced piston Mk 25 will produce flow rates of approx 300 SCFM while the diaphragm MK 16 will produce a flow rate of about 170 SCFM. A lower performance unbalanced piston reg like the MK2 will have a flow rate of 40-50 SCFM.
The tank valve also is a factor as some offer more restriction than others.
In any event the loss of a LP hose at the first stage is about as bad as it gets as there is much less flow restriction than with a freeflowing second stage. The only significant restriction present is the remaining threads from the fitting and the slightly smaller orifice that results.
Absolute IP will increase with depth, but the IP relevant to ambient pressure will remain the same so there no real increase in flow rate at depth with this type of failure. At extreme depths, the increased viscosity of the air under pressure should actually reduce the flow rate, but then at that extreme depth you'd be extra screwed anyway due to increased ascent time. As a practical matter, the IP the reg can effectively maintain in a high flow situation depends entirely on the flow rate of the first stage anyway.
But suffice to say that with a high performance first stage, a tank can empty much quicker than you would normally expect in that situation and far quicker than it will empty though a freeflowing second stage.