The reason is inherent losses of the engine. Internal combustion engines consume a lot of torque they produce for own operation - they need to compress the fuel-air mix, they need to eject the combustion products, inject fuel-air mix, run the cooling system, run the fuel pump, run the alternator for providing electricity for spark plugs, overcome mechanical friction of the gearbox and so on. This all takes from the energy/torque produced through combustion and decompression, and the energy intake/output characteristics are quite non-linear; at low RPM there's really little usable surplus output torque and increasing it requires increasing RPM - the growth of surplus soon outpaces growth of demand, and the engine reaches peak power (...then the losses begin catching up again, and the engine reaches peak RPM).
It's also connected with the combustion process: through pressing the accelerator you modify the fuel to air ratio in the mix, making it more energetic, but you can only go so far extracting certain amount of energy from a single combustion (single piston cycle) - further increasing of amount of fuel will just lead to ejecting unburnt fuel or flooding the spark plug, instead of increasing the amount of energy produced. Instead, you're better off increasing the number of combustion events per second - the frequency of piston cycles - the RPM of the engine. That way instead of minuscule amount of very fuel rich mix producing modest amount of energy, you have a much large amount of moderately fuel-rich mix producing a lot of energy.
In electric motor there's both very little of such 'self-maintenance' losses and the amount of output torque doesn't depend on some kind of cycles, like frequency of combustion events - energy input can be ramped all the way up, no problems like flooding the spark plugs with too much of it.
Now this is the torque-RPM relation. The resulting time-torque relation is simple: as a combustion engine starts, its RPM are low, low torque output. If you want to increase it, you need to "spin it up", increase RPM - and that takes torque, which you don't have in abundance yet - so the process takes time. In electric engines, you get full output immediately.