Deputy fuehrer was more like a US presidents Chief of Staff. It was not a Vice President equivalent. Hitler's designated successor was Herman Goering until just a few days before Hitler died.

Hess was deputy. Goering was leader of the Reichstag.

If Hitler would have died suddenly, in the manner you say, the world would be changed in every way from what we now know. Without Hitler there is no invasion of Poland-i mean who else in Germany really wanted to start another world war. The Third Reich begins to disintegrates , we can even expect a palace coup by the Werhmacht against the SS. Hitler held the whole thing together, including the populace, which saw him as "The chosen one" without him, The SS would try to hold it all together, but like i say, the Werhmact wouldn't listen to Himmler or Goering. Goering was greedy but soft, being the chancellor of Germany would have been enough for him, he wouldn't want or desire war with France.
The odds are we would have joined up with the Germans sooner or later to fight Stalin-another psychopath-us and the Germans would have been allies.
No Holocaust. The SS would be disbanded, and Himmler probably tried, and either hanged or put in mental hospital with the rest of his lunatics.
Goering, is definitely a lets go along and get along type of guy.
Hitler was the most evil man who ever lived, but he also was a genius, of the criminal type, a true mastermind, without him-the plans are off.

IF?? Every time someone said "IF" my dad would always say "If the dog didn't stop to sh#t, he would have won the race, lol

Since the invasion of Poland was planned jointly with the USSR, it's hard to see Hitler's successor/s (whoever he/they would be) calling it off. So war with Britain/France occurs as historical. A more interesting question is whether or not the new leadership later proceeds to make the two fatal strategic mistakes that Hitler made: invading the USSR and going to war with the USA.
It's also hard to see how Hitler's death puts a stop to the Holocaust, but maybe the European Jews and others are spared the worst of it.