Assuming you start counting from the point where your flow at the far end goes to zero and you are still pumping in at the same rate, my best guess is about 10 to 15 seconds to reach the dead head / no flow pressure for a pipe of about 300m.
You can work it out if you want using a transient simulation program, but how accurate do you want to be and why do you want to know?
The fluid will compress a little bit (Bulk Modulus) and the pipe will expand a little bit which is why the answer is >0, but it's much less than 1 minute.
You're probably looking at about 0.5% of the pipe volume to go from low pressure to dead head pressure.
But this is all very vague as you have not defined how much pressure is to be in excess of the steady state - 10 psi? 100?, 1000?, size or wall thickness of the pipe, Bulk modulus of the fluid, pump curve etc.
All of the above is based on this being 100% liquid. Throw any gas in there and the answer is different.
Remember - More details = better answers
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