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The Nazis expected an invasion for a while. They even talked about it in their propaganda and were relieved, when it happened. Goebbels wrote on th 16th of May in his diary, that they needed more propaganda and the Nazi party was supposed to strengthen the home front. They hoped that the invasion would be a chance to decide the war. After D-Day, the party wrote to their members in Berlin: "The party was right. The Führer is always right!" Of course, this excitement didn't last very long.

Fröhlich, Elke (Hg.): Die Tagebücher von Joseph Goebbels. München et al. 1993-2008.

State Archive Berlin: A Rep. 244-01, 89.

This seems like a good summation of the tactical aspects on the axis side.

The impression I got from the text is that the Germans were too preoccupied with their own blunders and red tape and the higher echelons seemed to underestimate the general scale of the invasion.

Rommel, upon taking the command found out that the defenses and fortifications had been greatly exaggerated by Goebbels and were nowhere near adequate. He believed the imminent invasion should be stopped at the beaches otherwise the allied logistics could easily overwhelm his forces. Von Rundstedt, Rommel's immediate superior believed the opposite and wanted to defend deep inland.

Von Schweppenburg who was in charge of the tanks in the area believed there wasn't anything they could do to stop an invasion and wanted to counter-attack once the beachhead was formed. This sounds foolish at first but it has some merits, Firstly the guessing game of the place of the invasion is eliminated as a factor, then an invading force is not very easy do defend - no time to dig in, lastly if he could close the gap between himself and the invading force he could eliminate air support for the allies. Anyways, he didn't get his way either.

Any armor movement in the area had to be authorized by Hitler personally, partly due to his megalomania, partly because of the severe fuel shortages. The Führer was asleep at the time of the invasion and Alfred Jodl refused to wake him up for several hours so at the end neither defense approach was implemented.

Great question!

Lets get started:

For starters it is useful to explore the tactical visions of what a landing would entail for the Reich, these visions varied.

Doenitz was of the opinion that "A maritime power which intends to undertake an invasion always retains the strategic and tactical advantage, since the choice of landing points remains in its hands. For a continental power which is called upon to defend its coastline it is therefore always difficult to decide which are the right places upon which to concentrate the main weight of defence; for to be equally strong all along the coastline is impossible." Doenitz was of the opinion that the Wehrmacht was wrong on their assessment that the Allies would land in the Eastern Channel close to the Somme area. The Navy, fortunately for the Allies, lost out to the army on the vision it had for the area of landing. The Navy believed that, for reasons of simple seamanship, the Allies would land in the Baie de la Seine and quickly capture Le Havre. The Navy, again to the good fortune of the Allies, lost out on all the major points of disagreement it had with the army. From the placement of the coastal batteries to the placement of minefields. Priority for the former being given to barrage fire rather than direct fire and placement to the east of Le Havre rather than in the Baie de la Seine. On the night in question, Doenitz recorded this in his personal diary: "I myself received news of the landing in the Baie de la Seine at about two o'clock in the morning. Between five and six o'clock Admiral Voss, the naval representative at Supreme Headquarters, telephoned to me and said that Headquarters hesitated to send all available divisions at once to the invasion area. I asked to be put through to General Jodl and told him that I thought this was the real invasion and that it seemed to me that our counter measures should be set in motion as a matter of great urgency. Jodl replied that the Commander-in-Chief, West, still had doubts on the subject. I flew at once to Headquarters." This shines a little light on the disfunction between the Navy and Army regarding the landing. It appeared to Doenitz that with the success of the landings, there was little question about the state of affairs the Reich found itself in, this was Doenitz's recorded opinion: "Once it had succeeded, however, we found ourselves faced with war on a number of land fronts. The war could no longer be won by force of arms. To make peace was not possible since the enemy would have none of it until Germany had been destroyed. For the fighting men there was nothing for it but to fight on."

Rommel and von Rundstedt had by now fallen squarely in the same camp as Doenitz. Following the landings both commanders had met and decided that since the odds were slim at best, the time had come to visit Supreme Headquarters and speak directly to Hitler. Now keep in mind that these are the two senior military commanders in the West. Their opinions were hard to discount. They met Hitler on june 17th, Rommel wanted to conduct the defence as he saw fit and turn back to his original idea, pulling his troops from the coast, regroup, and prepare for an offensive. Hitler told them that it would be quite unnecessary as new weapons would solve the Allied problem in the West soon enough. This wasn't enough for Rommel who argued with Hitler, saying that there was no way the war could be won. The Allies in the West had too many men, tanks and airplanes to be held back for too long, without a mobile defence it would be impossible and the East simply couldn't be held against the massive Soviet armies. This really got Hitler mad and he apparently had one of his major blowouts which really soured his relationship with Rommel, during said blowout, Hitler told Rommel "Don't you worry about the future course of the war, but rather about your own invasion front." The next meeting of the western command with Hitler and his staff in late June went worse. This time it was von Rundstedt who blew his top and gave a really stern, insulting, dressing down to Keitel which was followed by a phone call later that day from Keitel to Rundstedt who asked what could be done about the allies being rather successful in the West, which drew the answer "Make peace, you fools. What else can you do?", Rundstedt was removed from his post shortly afterwards.

It's important to remember that Hitler had taken control of when and why the Western command could move the armored reserves and that he had gone to sleep during the morning of the landings. It was Nazi thinking, the bullying and intimidation typical of the high command/Hitler's entourage during the late war that made waking him up a dicey prospect. Hitler had enablers around him and after D-Day, it only got worse with the removal of Rundstedt and the death of Rommel. What would happen, the reaction to the landings, would be a drawing back of Hitler around his inner-circle and a gradual move towards being far more paranoid... The power structure around Hitler discouraged dissenting opinions, by choice really, and with the landings Hitler would only grow more delusional. In large part because the most substantive change in the German High Command that isn't a physical landing of troops is the dynamic that came about after the attempt on Hitler's life. The bomb plot, if it hadn't been for the disastrous handling of the landings by Hitler and his close circle, might never have happened or may have happened later. Rommel is thought to have turned in favor of the assassination rather than forcing peace less than a month after the landing. The landings, if anything, made the nazis, or rather the nazis with a lot of power, far less rational.

Sources: The Memoirs of Field-Marshal Wilhelm Keitel edited by Gorlitz, Ten Years and Twenty Days: Memoirs of Grand Admiral Karl Doenitz by Karl Doenitz, D-Day by Anthony Beevor.

Piggybacking off OP's question, what was the Nazi/Axis reaction to Operation Bodyguard and the Allies' huge misdirection and misinformation?

I have a follow-up question of sorts: I remember reading years ago in a biography of Rommel that Rommel correctly worked out that the landing would be at Normandy, while others in the military bought the fake landing site the Allies feinted toward -- and Hitler decided to believe the others (due in part because he suspected Rommel of wanting to stage a coup).

The question is, simply, how accurate is all of that? And in general how concerned were Hitler and his cabinet (for lack of a better term) worried about staged coups by the military?