How long before an increase in synthroid (thyroxine) from 5 tabs of 25 mcg to 8 tabs each week takes full effect for hypothyroidism treatment?

Few weeks: It will take a few weeks to have a the full effect of a dose change of your synthroid (thyroxine).

6 weeks: Synthroid or levothyroxine does not have to be titrated like that as the half-life is 1 week and thus steady state typically takes 4-6 weeks to obtain. Generally, the most appropriate dose should be started and adjusted every 6-8 weeks according to bloodwork. A good estimate is 1.6 mcg/kg as a start.