Well, I'll preface this by correcting a few presumptions; the only mobile division in immediate striking-distance of the beachhead, and authorized to do so outside of the OKW giving the okay was the 21.Panzer. The 12.SS was released shortly after Hitler woke up - and he was not woken up during the night due to OKW responding cautiously to the confused reports coming in from the 7th Army. The entire division wouldn't be heading towards the beach-heads until almost 12 hours later as a result.

The 21.PanzerDivision's counterattack was delayed by the airpower running interdiction, but was able to commence before the end of D+0, though certainly not within the first 5 hours. The following quotes are lifted from an earlier post I made on the subject:

...throughout their long drive up the road to the battlefield that morning, even after the lifting of the cloud that hampered air operations for part of the morning, 21st Panzer suffered little material damage from Allied fighter-bombers^1

...

The last important action on the British left on 6 June was the battle against 21st Panzer's armored thrust, an action which went entirely the way of the invaders... The Germans recoiled westwards on meeting fierce fire. When at last they encountered the heavily-gunned Sherman Fireflies of the Staffordshires on Hill 61, the consequences for the Germans were devastating. 13 tanks were immediately destroyed. Only a handful of 21st Panzer's tanks...reached the surviving strongpoints of the 716th [German Statik] Division around Lion-sur-Mer. By a dramatic coincidence, only minutes after they did so, the fly-in began of 250 gliders of the 6th Airborne Division to a landing zone eastwards, around St. Aubin... ...by nightfall, they were [falling back towards Caen] with the support of 24 88mm guns. But they had lost 70 of the 124 tanks with which they had begun the day.^2

The 12.SS, by its own reports, was only able to conduct preliminary reconnaissance on D+0, with its Panzer-Aufklarung detachment being the only unit to arrive in the battlezone whole and ready. The rest of the division, released belatedly by OKW and facing the same incessant interdiction from allied air power, was either still marshaling or was en route. Regardless, they were able to counterattack after June 7th^3 against Canadians, an attack which ultimately failed with alarming losses despite its ferocity and competency. The Canadians likewise suffered heavy losses in men and material - essentially setting the tone for the Battle of Caen for the next 2 months.

As to the command paralysis, as already stated, OKW was sluggish in its response. More over, Divisional and Corps leaders - men on the spot who could've responded locally with rapidity - were mostly not even in the battle-zone when the invasion landings began. Most were away, participating in a war-game. Many had to run the gauntlet of paratroopers to regain contact with their units, and not all survived the journey. One of the first German losses on D-Day was indeed the General Commanding of the 91.Infanterie, ambushed while rushing back to his headquarters by wayward American paratroopers. The general confusion caused by the vertical envelopment cannot be over-stated; by accidental merit of their often wild misdrops the Airborne force, sizeable to begin with, deluded the Germans into thinking the numbers of Paratroopers involved were much greater than they were. Local infantry reserves were squandered on fruitless security sweeps in these precious first 5 hours of areas that only in reality contained small pockets of paratroopers, or none at all. Many more would not be able to secure de-trucking points or counterattack routes due to organized resistance.

Finally, organizing an operational counterattack in 5 hours is no mean feat - even for the relatively veteran HQs of Army Group B and the 7th Army. Especially if the invasion's exact landing position remains a point of contention. Skillful Allied deception, coupled with internal German political details^4 basically doomed the Germans to a flailing, lethargic response to the beachhead. To the credit of the German units on the spot however, many of the Allied objectives meant for Day-1 seizure would elude them for over a week, or in the case of Caen, a full two months.

^1: Max Hastings, Overlord: D-Day and the Battle for Normandy, page 112

^2: Ibid., pages 116-117

^3: Source Background states:

D-Day on 6 June had left the three British and Canadian beachheads reasonably secure, thanks to a slow German response and lower than anticipated casualties, yet the gap between the British 3rd Division and the 3rd Canadian Division was troublesome. There had been no counter-attack during the night of 6-7 June because the Germans were simply not ready for such an enterprise. The 7th Canadian Brigade was ordered to resume the advance at 0600 on the morning of D+1 (June 7) in order to gain their D-Day objective, a phase line code named OAK, running along the road Bayeux-Caen.

My emphasis.

^4: I'm referring here, primarily, to the famous Rommel-Runstedt dichotomy. The disagreement between the former and the latter was over how exactly to employ their armored reserves in the event of a descent. Rommel wanted to put all armor forward for immediate tactical counterattacks. Runstedt disagreed, citing enemy air and naval firepower -- certainly a legitimate concern given the Herman Goering division's counterattack having its back broken by naval firepower in Sicily a year earlier. He preferred to gradually scale back to a defensive-line in land, and perform operational counterattacks beyond the naval umbrella. Regardless of whom was ultimately right, OKW settled on an anemic compromise that did nothing but serve to further diffuse what was already an alarmingly insufficient reserve. Some units remained immediately forward, the 21.Panzer, while the rest remained picketed further back in general reserve.