First of all: this is definitely not an issue with LHR and BA. But AMS is a huge airport with plenty of runways and available slots.
It all comes down to one word: competition. KLM doesn’t like it that major competitors with a lower cost-base are now flooding AMS.
First the ME3 came, Easyjet appeared and now Ryanair. All are seen as unfair competition, although they are actually just “competition” ther is hardly anything unfair about it.
All the above mentioned airlines pay for the airport services and are subject to the EU and Dutch commercial codes.
Nevertheless KLM will survive in my opinion as they offer a very convenient feeder and long haul system. In the end they may even benefit from the likes of Easyjet and Ryanair as they will extend the feeder choices of travellers. For instance: KLM offers 26 destinations in the UK and Ireland with a single terminal convenient transfer to short and long haul destinations.
In stead of fighting the new kids on the block they should explore the commercial opportunities and start focussing on growing and selling again.
I would worry more about the French at Air France than about Ryanair or Easyjet. With the recently announced plan for lay offs at KLM, rumoured to be as much as 20% of the current 30.000 strong workforce, and nothing whatsoever at AF, the management at KLM has an even greater challenge.
And as for Willy: he would have every right to get nervous if Ryanair, Easyjet, Norwegian or any other LCC would sneak in at LHR. BA would have some real competition at their homebase for the first time.
BA management will experience what other airlines at airports with spare capacity have already encountered. A whole new game.
But basically: whether it is KLM at AMS or BA at LHR: they will cry foul every time a competitor enters what they believe to be their “nest”.
As others already said: they better grow up and fast.