The British giving pikes to the Home Guard was a desperate attempt to give them something. Against bolt action rifles or sub machine guns, hell even against pistols a small group of Home Guard with pikes were simply committing suicide. There was none of the training that made pre-gunpowder era pikes so effective, nor units of adequate size. Against a few parachutists, guarding prisoners, they might have some utility. When you look at some of the "weapons" the British cobbled together for the Home Guard, you realize that most of the Guardsmen were going to go in to combat on the basis of "you can always take one with you". Although the possibility of some Fallschirmjaeger laughing themselves to death seeing a squad of Home Guard advancing against them with pikes cannot be excluded.
To be fair, the numbers of Home Guard joining up was greatly in excess of what had been anticipated, resulting in a weapon shortage. As a result, it was decided that weapons would be allocated according to likely need, with areas considered closer to predicted landing areas getting priority, and areas far from likely landing areas being on the bottom of the heap. Thus Home Guard units in Kent and Sussex were quite well equipped (in particular, the Home Guard in the Dover area after Dunkirk was surprisingly well equipped. Apparently a number of weapons had been recovered, and the commander of the Home Guard unit, when instructed to return weapons recovered from the evacuation, replied in a well-known message that he would return every weapon his force found floating.) By contrast, Home Guard units in Wales were poorly equipped.
Of course, the limitation of the Fallschirmjaeger was that they dropped separately from much of their weaponry, and as experience in Crete demonstrated, they were vulnerable in the first moments of landing until they had actually found their weapons. Under such circumstances, if pike-armed Home Guard happen to be in the right place, the paratrooper may be at a tactical disadvantage.