It's surprising how much different the weather can be when your at the bottom of a valley as opposed to the top. I've not climbed skiddaw. I once set off on a walk in August from Rivington heading up to Winterhill. I had shorts on and a tee shirt but had packed cold weather gear. The walk through the gardens up to the pike was oppressive the heat was similar to as you described. I must of consumed a litre of water before i reached the tower.
Anyway within 5 mins of leaving the tower and heading out on to the moors towards the summit and masts the temperature plummeted to about 5 degrees and a cold wet fog descended on me , i then began to shiver so quickly threw my cold gear on. It was then that i realised why it was called Winterhill. I've experienced those conditions climbing peaks in the lakes but didn't realise these hills were similar. But most people do not realise that when venturing up there. They see the beautiful pictures people take on clear days from the summits and assume its just like a pleasant stroll in the countryside so venture out ill prepared and under equipped for a changing environment.