This sounds like a simple problem asking for a terrible solution...
What is the range of variability in the signal? If it were the exact same every time, you wouldn't need to ask, you could just use a timer.
What can you tell us about the source?
Why do you think you need to know the voltage (or time, or anything) so precisely?
~4V isn't "quite close to 0V". It's 20% of the full scale shown, more like a "generous fraction"...
Take second derivative, threshold the result.
First..?
Note that derivative is noise sensitive, so you're generally going to have a band limited derivative (i.e., it only acts like a derivative over some range of frequencies). This range depends on the characteristics of the source, and how fast your circuit needs to operate.
In addition to Marco's idea, the initial rate of increase shown is approximately 33mV/us, so to achieve the desired 1mV accuracy will need a fast sample and hold gated with a pulse of the order of ten nanoseconds.
Wait, who said anything about S&H?
If you mean, doing it digitally, yes you could do that, but if the given waveform is truly representative, you can get by with much less than that sample rate (about 20kS/s is minimal, by the looks of it).
Propagation delays become significant at this timescale so the signal to the sample and hold will need to be delayed in a transmission line, with an adjustable delay applied to the sampling pulse to 'tune' it to the point of inflection by matching the signal and sample pulse total delays. If this must run unattended, temperature compensation is likely to be required to keep the delays matched. Feed the sampled voltage to a 16 bit ADC converter which doesn't need to be very fast but does need good linearity and accuracy.
...And transmission lines and compensation oh my?
...Actually, if you're doing it analog, that's a good tip: using transmission line delays to account for comparator and S&H propagation. Triggering on the derivative change (assuming the dV/dt window is suitable -- we don't know yet) then picking the point "in the middle" will do the job.
To elaborate on that "slow sampling" point:
If it's a smooth ramp, you only need two points on it, before and after the slope change. It can be timed or sensed. If the subsequent time constant is consistent, then only two points are needed there, otherwise three (to solve for the curve's rate and magnitude). Solve the line equation for the pairs of points, and find the intersection.
If the source is noisy (and has more variation in the timing and rate and everything), more samples will be necessary. Some active analysis may also be required to determine where the slope changed. When that's found, do the same process, but use linear least squares interpolation to find the line equations and intersections.
This can be performed in digital or analog. The analog implementation might use a series of S&H's triggered by a sequential counter, then their outputs are compared (differentially) against a variable reference: a chain of resistors between two voltage sources (thus giving slope and offset for N points on a line). The values of those sources are wiggled (with some reference ripple) until the error signal (ideally, the RMS of differences, though max(abs(err)) could be used as well) becomes balanced. Lastly, some op-amps solve the equation of the lines (just a ratio of voltages) to find the intersection point.
Such an analog computational approach won't at all be very compact or expandable (you need N S&H's for N samples in the data set, plus muxes and other stuff if you want to do the computations serially, or N of those to do it all at once), but it would be capable of sufficient accuracy (the requested parts per thousand, without much trouble; pushing that into the ppm is probably wishful thinking though). More of a mental curiosity, if anything, but interesting anyway.
Tim
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