No it does not. Your billing cycle will always be the same.
It is recommended that you switch at the end of your billing cycle.
If you did switch part way in, your old plan and new plan will be prorated (21 days on the old plan and 10 days for the new plan). This means your old and new talk and data should be pro-rated as well (but Koodo doesnt currently prorate data).
As for the $20 data add-on, I am unsure. I believe it will be charged in full so you get to use the full data bucket in that add-on