How many soldiers that served in the Wehrmacht during WW2 held strong National Socialist convictions and ideals?

Before the war, or after? What you're asking for is an impossible number to quantify.

The bottom line remains that whether they were fervent or not, they were political tools and were quite aware of this fact. The "Hitler Oath" was made for the Wehrmacht just before it was formed from the Reichswehr. I'll pull from a recent answer, sources for the oath are within.

I swear by God this holy oath: that I will always be unconditionally obedient to Adolf Hitler, Leader of the German Reich and people, Supreme Commander of the Wehrmacht...

The year this oath is put into practice is 1934. Some sources put 1935, but regardless, its well before the actual start of WWII.

It still doesn't excuse any crimes they may have committed.

Indeed it doesn't.

The linked post, the sources I bring up also mention that military service at this time is very much built up to be an honorable calling - something it hadn't been in Germany since the conclusion of WWI. Men may have been conscripted, but the camaraderie and prestige the government connected with military service certainly would've embedded them with zeal - regardless of prior political leanings - before the start of WWII.

You may also find this post of great interest to you. I discuss and quote at length evidence of the heavy indoctrination of the German Wehrmacht. Again though, impossible to give you firm numbers of how many men were affected by it. Its clear based on the criminal actions of the Wehrmacht and its fighting spirit that this doctrinal bombardment took firm root.

As /u/BritainOpPlsNerf said, this is very difficult to quantify, especially since support for the state/nation and support for its ideology is a difficult distinction for a historian to make. German veterans postwar further muddied the waters by disavowing their involvement with the regime's crimes while accentuating their German patriotism. However, there is a lot of evidence that National Socialist ideology did penetrate multiple levels of the armed forces from flag officers to the ranks.

The Wehrmacht initially held the position that it was an apolitical institution and in the thirties, all NSDAP members had to resign from the party as a precondition of service. This restriction was waived or ignored as the military expanded and rendered the apolitical nature of the Wehrmacht fictive. There were multiple examples of NSDAP members serving within the armed forces during the war. Moreover, the oath of obedience further underscored the subordination of the military not to the state, but to Hitler himself. This is an important, if a bit subtle, distinction to a normal oath of loyalty for a military which tend to emphasize loyalty to governmental institutions. One of the key tenets of National Socialism was Führerprinzip, a highly personalized ideology that emphasized that Hitler stood above normal politics and law.

It is impossible to get the total numbers of NSDAP members that served during the war. There are a number of factors that make it difficult to give a firm projection of the numbers. Destruction of documentation of membership and other personal files during the war certainly has made this process difficult. The regional nature of German recruitment where individual units from the same geographic area (Wehrkreis) makes it difficult to extrapolate a general picture as NSDAP membership could vary widely from region. NSDAP membership also fluctuated somewhat during the war because the crisis in German manpower forced the Wehrmacht to draw from younger cohorts. Accelerated officer training programs of younger troops, many of whom came of age after 1933 and were thus more socialized to National Socialism. Studies of extant cadet school members mid- and late- war show that many of them held leadership positions within the Hitler Youth and SA. The various branches of of the Wehrmacht also quite different and attracted individuals from different parts of society and had different age cohorts. Hitler reportedly quipped that he possessed a reactionary army, a monarchist navy, and a National Socialist Luftwaffe.

In general though, the officer corps' NSDAP membership hovered at around 30%. This is not surprising given that the officer corps recruited largely from the middle class and this was a sector of society which the Third Reich had made a great deal of inroads in recruitment. An important aspect of an officer's career were also officer assessments for "National Socialist convictions" in which a qualifying officer would judge the officer's commitment to the state's ideology. Similarly, studies of representative frontline divisions show that a little less than a third of NCOs also had some involvement within NSDAP institutions. Both these groups, NCOs and officers, acted as important transmission belts for National Socialist ideology to the rest of the ranks. In some cases this meant that when directives came down the chain of command to underscore the racial nature of the war to the rank and file, which was quite common in the early stages of Barbarossa, officers and NCOs were quite willing to do so.

Apart from membership within the party, there were other avenues of NSDAP socialization that influenced the Wehrmacht as a whole. When the war turned against Germany, the military instituted Nationalsozialistischer Führungsoffizier (NSFO) to inculcate ideological mentality among the troops. The various organizations like the Nationalsozialistisches Kraftfahrkorps (NSKK) and Nationalsozialistisches Fliegerkorps (NSFK) (N.S. Motor and Flying Corps, respectively) were popular youth paramilitary organizations in the 1930s. These groups had a strong ideological component to them in addition to providing German youth opportunities to indulge in operating modern machinery like cars and gliders. The Reichsarbeitsdienst (RAD) was a compulsory youth labor service that grew out of Weimar organizations and aside from building bunkers, the RAD had a strong component of ideologically charged socialization. Although not run by the party, the RAD nonetheless was infused with major tenets of National Socialist ideology. RAD members were to be Aryan and there was a pedagogical component to the RAD to prepare German youth for military service. This meant that a large percentage of German youth had gone through some level of National Socialist organization, some voluntary, others compulsory by the outbreak of the war. As Germany's military fortunes fell, Germany began to scrape the bottom of its manpower reserves which meant more younger recruits. Soldiers from these younger cohorts had often came of age during the regime and thus were even more socialized with the state's ideals.

But to return to an earlier point, ideology and motivations are often very slippery things for historians. The compulsory nature of mass organizations like the RAD and HJ limited some of their power to indoctrinate, while voluntary organizations like the NSFK were appealing to youth more for their modernity than just ideology. And of course, the appeal of the Third Reich's ideology could be highly selective and conditional. The traction that National Socialism had among its population was hostage to its military fortunes. The regime had some of its greatest ideological successes when it grafted its ideology onto preexisting German social structures. In the case of the military, its exhalation of a strong masculinity fit well with the existing German military culture, but the social leveling of the Volksgemeinschaft made many older, conservative officers apprehensive.

Sources

Bartov, Omer. Hitler's Army: Soldiers, Nazis, and War in the Third Reich. New York: Oxford University Press, 1991.

_."Soldiers, Nazis, and War in the Third Reich." The Journal of Modern History (1991): 44-60.

Militärgeschichtliches Forschungsamt. Germany and the Second World War, Volume V/II: Organization and Mobilization in the German Sphere of Power, Wartime Administration, Economy, and Manpower Resources 1942-1944/5 Oxford: Clarendon Press, 1990.

_.Germany and the Second World War, Volume IX/I Politicization, Disintegration, and the Struggle for Survival. Oxford: Clarendon Press, 1990.