If you don't mind walking along a main road, then up the Cottage track to the A515 (0.5k), and then south along the main road for 1.5k, will get you to the Jug & Glass - good food and a pleasant atmosphere but a wee bit "restauranty". So around 2k altogether and actually the closest pub.
The walkers route to reach the Waterloo in Biggin, a proper drinkers pub which also does food: from the Cottage, go south along the Tissington trail for approx. 2.5k to the Heathcote road bridge crossing (just over 0.5k past Hartington signal box), go down the minor road into Heathcote (300m), turn left at the "roundabout" and carry on down the lane and along the footpath across the fields (approx. 1k), and you'll come out right by the Waterloo in Biggin. So around 3k altogether. By road it's a bit further but it's easy to find as Biggin is signposted from the A515, about 1k past the Newhaven turn, where the garage is.
As Paul says, the Royal Oak is now effectively a restaurant with very little space for drinkers. (The Bull i' t' Thorn, 4k along the A515, is now closed.) However the Duke of York, about 5k from the Cottage along the main road towards Buxton, is an excellent pub which also does good food. There is also the Bull's Head in Monyash, about 4k to drive.
So nothing really close by now but quite a few within reasonable distance. Or else BYO.