I made a mistake above. Wilhelm was considered as right-wing, not centrist candidate. I still think him becoming president is possible. Pod would be an earlier death of his father and is followed by his gradual realisation that the NSDAP is not monarchist at all. I made a short tl. As it is just a short side project it is not well researched, not terribly likely and not very detailed. Only a very rough draft to give an idea:
Winter 1929/30 After a short pneumonia Wilhelm II. dies. His funeral is not a big event, but causes some friction within Germany´s right wing. Within the NSDAP party members attending a memorial service are often criticized heavily, a discussion not going unnoticed. In a similar vein monarchist sentiments shortly rival the völkische ideology now dominating the DNVP and affiliated organizations to the disgruntlement of its proponents. On the party line it has no noticeable effects though.
1932 Hindenburg proves unwilling to candidate again, making the search for alternatives difficult. Only the communist early on decide on Ernst Thälmann for their candidate. On the far-right the NSDAP wants to field Adolf Hitler on their own. The other members of the Harzburger Front are unwilling though to submit to the leadership of their far-right partners. At the same time they realize the problem with splitting the far-right vote. Under these circumstances Wilhelm von Preußen is proposed as a joint candidate for the right, but the NSDAP immediately rejects.
Brüning, still unable to persuade Hindenburg to aim for a second term has not yet found an acceptable candidate who is also able to attract right-wing voters. Desperately he jumps at the proposal from the DNVP, although many moderate politicians have issues with Wilhelm well known far right political position. But after a number of personal talks between him and the party leaders even the SPD grudgingly gives him support.
This is exactly the opposite of the DNVPs intention, but since they made the proposal in the first place they are unable to back out. Still their support for the crown prince is lukewarm at best. Subsequently their vote is divided between Hitler and Wilhem von Preußen, while on the far-right Thälmann profits from the uneasiness the SPD members have with the Hohenzollern prince. He receives 26,2% of the votes, against Hitlers 35% and Wilhelms 38,6%. Gustav Winter gives up his candidacy with only 0,3%
Smelling victory the NSDAP launches in the short period to the second ballot a furious campaign and attacks the crown prince directly, which has a serious impact on his view upon the party and destroys a previously decent personal relationship with Hitler. But the one loosing the most in the second round is Thälmann. Not only many dissatisfied SPD voters shift back to the mainstream candidate out of fear of Hitler, a number of communist core voters also go over to the more promising “working-class candidate” Hitler. In the end Thälmann falls back to 10,1% compared to 38,8 for Hitler and 48,6 for Wilhelm.
Only about 3 month later, in July 32 the new president is deeply enraged by the results of the conference of Lausanne and withdraws his support for the government Brüning, which resigns the next day. Initial talks to form a government fail, but the NSDAP shows willingness to enter a coalition in exchange for new elections. In those the far right fails to achieve an own majority although the NSDAP doubles its votes, becoming the strongest party. After talks to gain the support of the Zentrum fail, for which the blame is placed mostly on Hitler, Wilhelm, urged by acting minister of defense Kurt von Schleicher tasks the largely unknown Papen with forming a government.
The new government is composed from politically unremarkable figures and seen as a transitory one. It rules only via Notverordnung, but the measures indeed contribute to an improvement of the situation. Its restrain towards the unstable situation in Prussia is noted as well (although it was in truth the president prevented a deposition of the provisional government, fearing the legal repercussions of a coup by the federal government). Still the government has to dissolve the parliament already on its first meeting, only legally preceding a motion of non-confidence by KPD and NSDAP. In the new elections in the December of 32 DNVP and to a lesser degree DVP, the only supporters of Papen, gain some votes as does the KPD while all other parties loose, especially the NSDAP. With these results NSDAP and KPD narrowly loose their blocking majority.
Again Papen tries to gain NSDAP support, only to break talks due to their still excessive demands. An attempt to split the NSDAP also fails. But by now the moderate parties are at least willing to tolerate a government led by Papen with personnel changes and under certain conditions. Neither are really liked by the president, chancellor or the eminence gris, Kurt von Schleicher. Especially the return of Wilhelm Groener irks the latter to no end. They accept nonetheless and a government of national unity is formed. They continue to rule by Notstandsverordnung, but the parliament at least prevents a motion of non-confidence.
The government continues to expand its investments into the infrastructure and with it the Öffa-bills financing them. Guided by the hope to gain the concession to rearm moderately at the Geneva conference more and more is invested in the preparation of said move. Unfortunately the talks break down in October 33 when France again vetoes a British compromise offer. Germany now sure that France won´t negotiate honestly leaves the conference.
The same month the NSDAP has to declare bankruptcy after business sponsoring dried up and more and more members leave. Shortly thereafter allegations of coup plans lead to a renewed prohibition of SA and SS.
Accepting the slow, but high-quality rearmament of the German armed forces as a fact Britain tries at least to keep it from endangering its own security. Therefore in 34 a naval treaty is signed, restricting Germany in essence to a third of the Royal navy. A few month later a similar treaty regulates the aerial rearmament. In exchange for strict restrictions on the production of heavy bombers Germany is granted in essence a parity of force. Much to French displeasure the expansion of German ground forces is essentially accepted without restrictions.
In January 35 finally the unstable coalition collapses and new elections are scheduled. With the pressure of the economic crisis much reduced the never popular government is felt more of a burden and old disagreements resurge. For the NSDAP the elections are a disaster. They loose more than three quarters of their seats with a result slightly below 8%. With that they fall behind even the KPD which also losses and falls back to 10%. The great winners are the SPD, once again biggest party and the DNVP, to which cancellor and president are commonly counted, though not being members of the party. The Zentrum wins as well and the DVP almost doubles its results and thus reappears as a factor on the political stage.
1936 the Olympic games in German are a huge success and improve the German standing abroad greatly. The same year a minor diplomatic scandal happens when the president shows some sympathy for Mussolinis war in Ethiopia. Comparisons with his father are drawn, but the uproar quickly ends as political actions in support of Italy do not follow.
1938 the Government of Danzig calls for a reunification with Germany and prepares a plebiscite. The Polish government immediately protests and threatens with a military intervention. For some time Germany and Poland seem at the brink of war, causing the German foreign minister to sign a secret treaty with the USSR dividing eastern Europe in zones of interest. But without external support, not even by France the Polish finally give in and agree with the provision that they retain their base on the Westerplatte.
With the success of the plebiscite the Austro-fascists loose their grip of power, already shaky because the economic recovery is slower than in the bigger neighbor. Initially all western powers are opposed to the idea. Only the Soviet occupation of the Baltics, as agreed in the secret treaty with Germany, softens the opposition. In the end a plebiscite is allowed by France and Great Britain in exchange for an armament treaty restricting Germany to equality with France, where the treaties with Britain limit them not further already, and even that only a decade later. As Germany is financially not really able to continue the rearmament they happily agree.
The scheduled presidential elections 1939 follow in the aftermath of the Anschluss and the crown prince candidates once more. Around the same time the demand is raised to reinstate monarchy and crown him as emperor...