I have travelled in my Alto car from Lahore to Skardu many times. Last time I travelled on KKH in April this year.
From Lahore to Besham, it takes almost 10 hours (non-stop). Four hours on the motorway up to the Burhan Interchange. Six hours after leaving the motorway. However, you can save time if you cross Abbottabad and Mansehra early in the morning as I have always driven through these cities during the rush hour.
From Besham to Chillas, it usually takes five-and-a-half hours. But due to bad road condition from Pattan to Dassu, it will take almost 7 hours.
From Chillas to the Raikot Bridge, it takes almost one-and-a-half hour.
In mid August, you can take Naran-Babusar-Chillas route. It is scenic, shorter and will take less time as compare to Besham-Chillas route.
Lastly, try to add a day or two if you really want to go to Fairy Meadows.
