After scanning a few planets in a star system, the Reaper bar fills up and they start coming after me. When will that bar resets so I can have access to the system again?
You have to do one mission, and all systems get their reaper awareness meter reset.
To make the most out of this you should visit all systems you haven't explored to 100% yet before going onto a mission, that way you get the maximum number of tries overall.
There is also the trusty old save and reload method here, just save before scanning and then scan down everything in one system. Let yourself be eaten by the reapers, reload and only scan exactly where you know the signatures are. That way you often don't alert the reapers at all, or if you do you're already close to the signature and can just pick up whatever there is and quickly get out of the system. While you're in orbit around a planet the star system map is paused, so you can even do that with the reapers on your trail.
To find the signatures that are not on planets without alerting the reapers you can also just fly around in space a bit and wait until you hear the sound effect you get when you fly near a planet. If you get that effect while not near a planet it means that there is a scannable signature nearby. This can reduce the number of scans you need and help you avoid alerting the Reapers.
It is very very easy to continue scanning a system even if the Reapers are alerted. I considered them a minor inconvenience, at most. Go in and scan until you find stuff, then if/when the Reapers come, flee the system before they get you. Return right away, flying directly to whatever assets you found but didn't pick up yet, or directly to wherever you haven't scanned. Rinse and repeat whenever a Reaper gets close. They are at their slowest when they first enter the system each time, giving you time to move around a bit.
Just because Reapers are there doesn't mean you don't have access to everything in the system.
Also, remember the trick that whatever direction you ship is facing on the map will be the way you enter the system....so if my ship is pointing down on the cluster map, it will enter from the "north" side of the system, still facing down. If my ship is facing right on the cluster map, it will enter from the "west" side of the system, ready to travel right. This little trick can save you time when outrunning Reapers to make direct paths for scanned items or to exit via Mass Relay or refuel.