I know large numbers of Jews served in all of these armies, but how much prestige and authority could they earn? What was the furthest that Jews could advance in them?
I'm going to discuss the situation in the German Army.
First off, some organisational context is needed, because the Imperial German Army had a uniquely convoluted structure. There was, in fact, no single German Army and no Imperial War Ministry. Instead, the Kingdoms of Prussia, Bavaria, Saxony, and Wurttemberg each maintained their own separate armies and ministries of war. The Saxon and Wurttemberg armies were largely annexes of the Prussian Army, but while the Bavarians had to use the same equipment and follow the same organisation, they had considerable administrative autonomy.
At the outbreak of the war there were no Jewish officers in the Prussian Army. No legal barriers prevented Jews from being commissioned and Article 12 of the Prussian Constitution guaranteed the freedom of religion, but the army simply rejected Jewish candidates. Some converts from Jewish families did become officers, although they continued to face discrimination.
The situation was slightly better in Bavaria. There were no active-duty Jewish officers, but there were approximately 100 officers of the reserves. Perhaps unsurprisingly the transport corps was the most willing to accept Jewish officers, 17.5% of reserve officers in the Bavarian transport corps were Jewish.
Once the First World War broke out Jews almost immediately began to receive reserve commissions. This was partially due to the Prussian Army’s policy of promoting reservist NCOs based on educational attainment; university educated Jewish reservists who would have been officers if not for their religion became NCOs instead. By 1915 the Prussian Army had commissioned around 500 Jewish officers and by 1916 it was 2000.
To the best of my knowledge no German reserve officers were promoted above the rank of captain; several Jewish officers reached this rank, perhaps most notably Bernhard Weiss. Although Weiss was born in Berlin, he chose to do his military service in the Bavarian Army and received a reserve commission before the war. During the Weimar Republic he became deputy head of the Berlin Police and was a notable opponent of the Nazi’s rise to power.
I’m not as familiar with the situation in the Austro-Hungarian Empire, but there were considerably more Jewish officers in service. One Hungarian convert to Christianity, Samu Hazai, rose to the rank of Colonel General during the First World War.
Sources
There where around 100.000 Jews in the German army. 10.000 of them volunteer.
The "Reichsverfassung" from 1871 made no difference for any religions. But Jews where banned from high offices around Germany. Only later, on the 4 August 1914, after the German Emperor Wilhelm II made following Kings speech:
„Ich kenne keine Parteien mehr, ich kenne nur noch Deutsche!“
„I don't know any parties, i know only German's!“
There was a slight shift in the army, and some Jews where getting non-commissioned officer ranks. For the Reserve, they could get lieutenant.
Alter on there was a growing antisemitism, that the Jews where weak and draft dodgers, the War-minister Adolf Wild von Hohenborn made a "Judenzählung" (Jews count) on the first Oktober 1916. The result was, that the Jews where equal in ratio to non Jews, also in the time spend on field/battle, injury's, time spend to recover.
But there was still a high antisemitism in the officer ranks. As a Jew, you will get easy demoted and harder promoted.
One of the notable Jewish officer was Hugo Gutmann. He was the officer who award Adolf Hitler the Iron Cross Class I. Gutmann was promotet after he captured 12 soldiers only with a pistol - what Hitler later claimed for himself. Gutmann got already the cross at this time, so he was honor in all regards.
In the "R.M.A. 22. November 1907 - Aid 10010 - " the standard for new marine Officers, stands that an officer "must be christian". Equal passenges can be fund the one for the Army.
In an correspondence between the Major and vice general staff of the army Joachim, and the mother of soldier of the 5th. reserve artillery regiment. We can find that her son got the Hessian Cross of Valor, and the Iron Cross Second Class, and was 5 times dismissed of an promotion. The only way for her son, to get a promotion they need to seek an audience with the emperor.
So over all Jews can only get non-commissioned officers, the only why for them to get higher ranks was from the head of staff or from emperor himself. What no Soldier was granted, all papers where dismissed, and not one reached the head of Army or the Emperor.
But Jews could get all kinds of honor, even the iron cross class II with black and white band.
Be for the war, 1880, there where around 25.000 Jews in the German Army. There was not one officer at any rank. From 1200-1500 Jews, who got baptized in the army, reached 300 the rank of an lieutenant and higher.