Most objects in Illustrator are unaware of their surrounding objects. There is no way to automatically move any object based on the stroke size of any surrounding object.
You're best bet is to manually adjust any overlapping you do not want or to select all and use the Scale Tool
or Object > Transform > Scale
menu item, ensuring Scale Strokes & Effects
is checked on the Transform Panel
. If you scale the strokes rather than increasing their weights manually you'll get thicker strokes and the distance between the strokes will also scale simultaneously. It is simply more difficult to target a specific stroke weight when using the scale method.