Souleymane Coulibaly opened the scoring and after Nuneaten had equalised Williams pounced at the start of the second-half after the visiting ‘keeper had spilled a Coulibaly shot.
An own goal following an Andrew Fox corner, a Nicholson tap-in after another new signing Adil Nabi had struck the crossbar, a superb 30-yard drive from Jonathan Edwards and a Fox goal following a fine pass from another newcomer Callum Chettle completed the Posh scoring.
Posh goalkeeper Dion-Curtis Henry made a couple of fine saves.
Striker Williams and midfielder Chettle joined Posh from Nuneaton last week.
Posh: Henry, Adebayo-Rowling, Goode, Trialist, Fox, Nicholson, Nabi, Chettle, Coulibaly (sub Edwards), Williams, Da Silva Lopes (sub Lee).