There is a way to change the width (almost) on the fly but I'm not sure it works how the developers intended!
I'm pretty sure what was intended (and I'm also pretty sure that is how it used to work) is that you simply press the + or the - key to change the width as you are routing but it's got broken in recent updates.
This is how it works now:
Route.
Stop the route.
Change the routing width in the right hand panel.
Click back on the last point you routed.
Start routing again from that point **but do not click to create the next vertex.**
Then either:
Clicking the minus (underscore) key now switches to the new routing width minus 1 increment lower in size and then press the plus key to go back up to the desired new width;
or:
Clicking the plus (equals) key now switches to the new routing width plus 1 increment higher in size and then press the minus key to go back down to the desired new width.
Chrome
80.0.3987.163
Ubuntu
EasyEDA
6.3.53