I have a cistern with an indoor water pump. I recently moved just my hot water line going to my upstairs (2nd floor) and connected the pipes with push fit connections. Since then I now have terrible water pressure on all my faucets (on both the first and second floors) even with the cold water line. What happened and How do I correct this?

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connected the pipes with push fit connections

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The use of push fittings not a factor here. No should moving the water heater affect cold water pressure.
Most likely moving the water heater disturbed sediment in the bottom of the water heater. How long had it been since you flushed the water heater? Note "flushed" not drained the water heater.
Sediment from the water heater has probably blocked aerators in faucets. Remove and check volume, clean or replace aerators.

Agree with Harold. Question: Does tub spout have same problem with water flow ?

At this point, you should remove all aerators ( the removable tip on end of each spout ) and clean them. Also, remove shower heads and clean flow restrictions and screens. Flow restrictions and debris screens are installed on the inside of the threaded part of shower head.

Hope that helps

Milo