As it says on the tin, could Prussia tip the balance in order to beat napolion at his finest or would they be to slow and we gust combine the fourth coalition into the third.
IIRC, Prussian envoy carrying ultimatum to Nappy’s arrived at Austrlitz almost immediately after the battle and had to deliver an improvise acongratulation to which Nappy made an ironic remark. So Prussia was ready, just too slow.
Now, it is an interesting question if a joined Austrian-Russian-Prussian army would defet Napoleon in a field battle at this specific time. I’m intentionally omitting the problems related to them getting together and considering the best (for the allies) scenario. While the purely numeric odds would be better for the allies, they are no guaranteeing the victory:
1. Clausewitz, who had both the first hand knowledge and unquestionable analytic abilities, left rather disparaging description of the Prussian army of that time from the top to bottom. Enough to say that the main Prussian army was defeated by a single Davout Corp at Auerstedt. Its commanders had been brave but they stuck to the old notions of the linear tactics. The troops were well drilled ... on paradeground. The “spirit” was strictly offensive (in a disregard of the numbers, terrain, threat to the flanks and other ‘trifles’). Even putting infantry into the squares against cavalry charge was rather a theory than accepted practice (Clausewitz managed to do that after Jena for the battalion in which he served but even then there was a problem with forcing soldiers not too shoot too early). So, if they are joining the allies, their proposals would be probably along the lines of “attack, attack, attack), further strengthening already prevailing mood.
2. There could be tensions regarding the chain of command. In OTL the battlefield commander was Russian (who was mostly interested in not taking any responsibility and preferably not fighting a battle) with the “brain” provided by an educated Austrian officer (who already had a record of leading the allies into almost hopeless situation in the Alpes). Now, the Duke of Brunswick would outrank Kutuzov in a military rank, military record (not too difficult) and social position (ditto for Prince Hohenloe). So most probably this relic of the gone age would be placed in charge making an overall situation even worse (at that point even Alexander could be pissed off with a situation in which he can’t get credit for a victory). In Austerlitz there was enough of confusion even with two participants and adding the third (with overinflated ago) would not improve things.
3. There was no pressing need for Nappy to fight a decisive battle specifically at Austerlitz and as soon as the allies are separated due to the logistical needs, he can beat them one by one. Taking into an account that none of the allies was on the same level as in 1814 while Napoleon still had all his narshals (especially Lances) and the troops arguably at their best level or close to it, there is a good chance for him to win a battle even against a joined army of all three.