There is no 'order', only verbal agreement that they will come to each other's aid in case they run into Napoleon's army. Wellington rode to Blücher and promised to aid him at Ligny but he didn't which resulted in Prussian defeat at Battle of Ligny. Blücher himself was incapacitated. Chief of Staff Gneisenau didn't want to come to Wellington's aid because he didn't trust the British. Had Blücher not insisted marching onto Waterloo, Wellington would be crush like Alexander in Austerlitz.
Wellington couldn't come to Blücher's aid as Napoleon blocked him with his left wing under Ney at Quatre Bras.
Your argument seems to be coming from the historical revisionism that Hofschröer puts forward, about how Waterloo was somehow a German victory and Wellington simply took the credit.
It was neither, it was a coalition victory by both Wellington and Blücher in close cooperation. Close cooperation by the standards of the day that is, in personal meetings and messages conveyed on horse-back.
Neither knew from which direction Napoleon would attack, and when he did he took both Blücher and Wellington by surprise. Moving from the south-west he defeated Blücher at Ligny, whilst pushing his left wing under Ney to block Wellington to the north at Quatre Bras. Arguably Napoleon's biggest mistake was not to make sure he had decisively beaten Blücher at Ligny.
Blücher withdrew to reorganise, and Wellington pulled back north along the Brussels road to Waterloo. Crucially Blücher also withdrew northwards, parallel to the French advance, and Wellington was well aware of this, the French not.
The rest is history. Wellington's army held the field all day, the Prussians arrived on the French right in the late afternoon, and with Napoleon's attacks exhausted, the French army routed.
To quote Wellington himself written immediately after the battle "I should not do justice to my own feelings, or to Marshal Blücher and the Prussian army, if I did not attribute the successful result of this arduous day to the cordial and timely assistance I received from them"
Whether or not Wellington promised to come to Blücher's aid at Ligny is debatable. Hofschröer claims he did. You can't call it revisionism because he was simply stating his view. Ask 10 people today who defeated Napoleon at Waterloo, 9 wouldn't even know Prussians were there.
Wellington has always reserved the option to retreat from the sea. That's why he left 10000+ reserve to cover the retreat path in case things go wrong at Waterloo. From Prussian point of view 1) they just lost a battle badly at Ligny 2) Wellington said he'd come but he didn't 3) Wellington could escape from sea in which case Prussians will surely be crushed facing French alone. Wellington does deserve credit but I would say Blücher's courage to trust his teammate and meet Wellington at Waterloo is more respectable. It's not a overstatement that it's a German victory.
Napoleon's biggest mistake is giving his marshals wrong tasks. He left his best marshal Davout in Paris while hoping Grouchy who just got promoted marshal to carry the weight to chase down Blücher. Grouchy is OK with carrying out orders but not so great at reacting to situation. Despite Gérard repeated asking to 'march to the sound of the guns', he decided to follow previous order and chase Prussians in the wrong direction. Although he didn't have too much of a choice, Davout is the only one whose loyalty is questionable. He didn't want what happened at Fontainebleau to repeat so he had to keep Davout at home. And then he used Soult for staff work who did a very sloppy job. The lack of information and misinformation may have well cost the battle.
