Oooh, now this is a fun question! Requisite disclaimer: do not try this at home, will most likely cause boom.
First, I'm pretty sure that water heaters never turn on both elements simultaneously. Even with a malfunctioning water heater, I'd imagine this would still be unlikely/impossible, as it would need to draw 4500x2 watts which would pop the breaker
Okay, then, I think it's unlikely the water heater is heating all 40 gallons from 70F to 120F in 3 min:
Multiplying 40 gallons by (50F/3 min) by the specific heat of water gives 98kW, which is ~20x more than what your water heater is rated for
If, instead, we assume it's only heating the ~2gpm for your faucet, the same calculation (2gpm * 50F * specific heat) gives 1500W. This actually matches nicely: if you used the full 6gpm into the water heater, you get exactly 4500W
The last calculation shows that when pumping out less than ~6gpm, some of the heat energy is going towards heating the water in the tank. For one faucet, 1500W is going to heating the 2gpm leaving the water heater and 3000W is going to raising the temperature of the water in the tank
However, even that last calculation is probably underestimating the heat going into the water tank. That's because it's likely not heating tap water 50F (from 70F to 120F); since water heaters have great insulation, the water tank is probably still pretty warm from the last time you ran it. If the tank is already at, say, 95F, then it only needs 750W to heat the outgoing water to 120F and 3750W are going to heating the water tank
Then, I went down a long rabbit hole: clearly, this is both heating the water tank and having water advected through. This is best modeled with an advection-diffusion equation:
We'd also need a source term for the heating element F(x,t); we can probably represent the source as a Dirac delta function in space. Luckily, this is a linear PDE. It's not hard to find the Green's function for F(x,t) = M δ(t-s) in time (e.g. boost to a co-moving reference frame):
Then, convolving this in the t-direction against a constant function gives a difference of error functions, which makes sense because the expanding region surrounding the element will warm followed by a strong temperature gradient which is advecting downstream but we need to make sure M has the right units coming from the 4500W and...
And I realized that was entirely unnecessary.
If we're just trying to determine how long you can operate the water heater without it overheating, we can ignore the outflow (since that will just extend the time slightly, so ignoring it is conservative). Then, going back to the original calculation, dumping 4500W into a 40gal tank should increase its temperature at a rate of about 0.8F/min (call it 1F/min)
So, the real question is: what's the current temperature of the water in there? It's hard to tell from the tap because it's likely mixing some cold water in. If it's a 50/50 mixture with "cold" 70F water, then that 120F could actually be 170F from the water heater. However, if it really has cooled off to 70F (again, big if!), then 1F/min implies you should have ~1.5hr before you start approaching 200F
Now, I personally wouldn't bet on that.
A far safer test would be:
Turn off the cold water line to a sink so you can get a measure of the actual hot water temperature
Open all the faucets/showerheads on hot so you're pumping out at least 6gpm (if that's the water heater's inflow rate, you won't be able to pump out any faster than that)
Grab a thermometer and watch the temperature of the hot water from the sink in step 1
Time how long you have until it gets to, say 170F
Effectively, instead of trying to theoretically calculate it, just measure it a la 10th grade physics class.
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