what his rule would look like?
Heydrich in charge would be a quite interesting time as he is the far opposite of Goebbels. Joseph was a chaotic force who rallied popular fervor, Heydrich acted behind the law as head of a Police State with complete indifference towards the people.
For starters, Heydrich likely won't accept a peace in the Ural War that is not his terms. He will mobilize the Waffen-SSF to the Frontlines as the self-sabotage scheme is over once Wegener and Hess are out. This increases the Russians attacks through Partisans but allows him to push back in the Orenburg Front, supporting Wenck against the Russian offensive there. This likely means a more favorable peace where the Germans keep Orenburg and, with it, the main Railway connection that goes around the Ural Mountains in Russia is cut off. That doesn't stop the Russians in the long term but strengthens Heydrich's position at home.
The next step is dealing with the agitated crowds of fanatics at home, I see Heydrich making a Pact with Speer and the Corporate sectors to stamp down a popular upheaval headed by Goebbels. Speer had a great working relationship with the SS during the war and many of Speer's sponsors were parts of the circle of friends of the Reichsführer-SS. With his control of the SSK (a merge of the SS and Police) and the RSHA, eliminating Goebbels would be easy. First by publishing his affairs and showing him mingling with the "untermenschen" will take the wind of the sails, it will take out the support of the fanatics who won't side with a man who sleeps with Czech women. After that, killing him would be as easy as a plane crash.
I see him being no less ruthless when in charge of the rest, Darnand will remain in charge in France and the pervasive influence of the SSK would extend to the paramilitaries of other countries such as the Blackshirts in Italy. When Almirante comes up calling for reforms in the Pakt, he won't be lasting for long, a scheme similar to the one made by Goebbels would happen, but I see measures taken to minimize the risk of the Italian atomic arsenal.
When it comes to the Church, Heydrich is neither a man who has personal occultism convictions like Himmler, nor a fanatic like Wegener. He will pursue a normalization with the Church on paper by reducing the pressure, but if Stephen X starts to act, he will send Skorzeny to the Vatican. But I'm quite sure that if the RSHA is in a better shape without purges, the Pope won't make it out of Europe alive. A more tame man, less overtly Pro-German, is put in charge of Rome after Stephen has a tragic cardiac arrest.
In regards to Israel, the Syrians still will receive German support, same with Nasser. But I also see a greater sponsorship of subversive movements led by Palestinean groups in Israel, likely with the sponsorship of terrorist organizations to destabilize the Jewish State. Once Heydrich believes the Israelis weakened enough, he will strike, but he won't be as much of a showman like Goebbels who made it as obvious as possible that he was going to march in troops to exterminate millions. So there would be less German direct action and more action through Arab and Syrian troops doing the heavy lifting. Israel won't have as much international support without Goebbels being the spokesman of the Apocalypse, but they have a better chance fighting Nasser and Saadeh with the Germans probably just sending a couple SSK divisions.
Heydrich's Germany is a security State led by an oligarchical alliance of powerful industrial interests exploiting the Continent with the Security Services. It's a place where the Wehrmacht is slowly replaced by the almighty SSK, where the Linz Pakt is unified through the SS rhetoric of a Pan-Europa by having the youth of these nations brought into the tutelage of the SSK and taught from birth that they were all the face of a New Europe, where the benevolent leadership of the Reich is exalted. It's something Schweizer's France does under Goebbels, but imagine that instead of the Volkssturm in Germany, the "energy" of the youth is in these rallies for a Pan-European unity against the Judeo-Capitalists in the West and the Asiatic Hordes in the East, a bastion of civilization and prosperity free from the materialism of the United States or the backwardness of the Russians.
More dangerous would be the German global influence because of the deep connections between the SSK and the Black Market, you will see drugs, weapons, even humans being trafficked out of Europe and into many nations, especially the United States. Without it's purges, the RSHA would be more competent in the Espionage wars against the Americans and start sponsoring movements to bring down the Free French Empire in Africa or the British, only to facilitate an eventual colonial expansion in the Continent by the Italians and French who will continue to employ their neo-colonial policies. You can expect a lot of support to places such as South Africa.
Heydrich either will be assassinated due to how careless he is of his own security, or he will be living a long time in charge because he will never be overthrown from the inside. It's likely he would, like Goebbels, be interested in setting himself up as Hitler's successor, more as a way to compensate his personal reservations towards the public. Which means Klara will still be put under watch and, eventually, marry either one of his sons or a grandson. It wasn't just Goebbels who had an obsession with blood and, considering Heydrich is self-conscious about his supposed Jewish heritage (one of the likely reasons why he overcompensated by going into daring dangers such as being a wartime pilot), he will try to "redeem" his lineage by marrying it off with Hitler's.
