Recovery time depends on how long he's been ill without treatment prior to diagnosis. It can take a year to a year and a half; with gradual improvements during that time.
They have to up the dose slowly at first (they will give him the first set dose, test his blood after 6 to 8 weeks and then increasE if necessary). While he's ramping up the dose in this way he may have days where he feels worse not better, especially in the first couple of weeks of a dose change.
You also need to get vitamin levels checked - others can advise on this better than me. Mainly it's foliate, ferritin and D; these need to be in the upper end of the range (not just in the normal range) to allow his body to metabolise the medication properly.
Sorry to be the bearer of bad news; thyroid is not a quick fix condition but the good news is that you do get better in the end and it is very treatable.