The reverse recovery of your diodes does influence the power dissipation in your application, but in this case is extremely unlikely to be the major cause of power/efficiency loss.
You could start by comparing a fast rectifier to you current rectifier.
The VUO52-16NO1 and a much faster VUE130-12NO7 are a good comparison.
While you gain lower switching losses in the VUE130- the power loss is actually dominated by the difference in Vf in your application (which I assume is a car type alternator). The VUO52- has a Vf of only 1.4 V for a total bridge loss of about 160 W at 60 A, and the VUE130- has a Vf of 2.7 V for a total loss of about 320 W at 60 A.
The reverse recovery losses for the VUO52- is unlikely to be more than about 16 W. Because you have a large output filter capacitor the diodes go into reverse recovery just after the peak of each phase voltage, but If has already dropped to close to zero dropping the stored charge considerably. Yes, there will be a reverse current flow, but insignificant compared to If max.
This might be worthwhile reading for you.
You could use fast and low Vf Schottky diodes to reduce the voltage losses.
For example the APT60S20B or the VS-100BGQ100 (more appropriate because it can be screwed down) would more than half your power loss.
The best way to reduce both switching and forward power loss would be to use a half or full synchronous rectifier of course. More complex electronics, but power losses in the rectifier can be reduced to just a few watts with fully synchronous.
SMART and Synch Rectifiers
If you look up SMART rectifier controls (such as this) you will see ways to use FETs as ideal rectifiers. There are lots of these controllers but be careful that you understand that some of these devices only operate at low frequencies, so may not be suitable.
One simple way to implement synchronous rectification is to sense the current in the lower diode of each pair and use a FET on the upper diode. This is simple and reduces your power loss in half. You could implement this with the 3 phase rectifier you already have.
Then you can get creative and build your own microprocessor solution. However if you aren't at that level you could consider using an open source ESC controller as a synchronous rectifier ....here is a demo of that on YouTube. Here the user is simply using the ESC32 as-is, but you could make it automatically track speed by altering the firmware. Again, depends on you capabilities.
The charge storage of a junction diode is a bit like having a tiny battery in parallel with it. During forward conduction, it 'charges up'.
If the voltage swings abruptly negative, then that charge must flow out again through the terminals. This is the reverse recovery current.
'Fast' diodes are engineered to have less charge storage, and/or faster loss of the charge when the diode is no longer in forward conduction. Once the voltage has dropped to zero, this charge storage 'battery' starts running down. Charges diffuse within the junction, and doping with certain materials, gold for instance, can increase the rate at which charges recombine. If the voltage hangs around near zero for long enough for all the charge to have dissipated this way, then when the voltage eventually goes negative, there's no charge left in the junction to drive a reverse recovery current.
Best Answer
Most of your questions can be answered "yes", but be careful with Schottky vs. (ultra) fast diodes:
"Can I just buy the higher voltage and current rated Schottky diode and sub them for the lower rated ones?" Should Work.
"Is the main reason for getting the exact match the cost and ability to fit?" Some issues like the diode's capacitance have an influence on switching speed and, as a consequence, switching losses, i.e. heat generated in the diode. Try to find at least a similar device. Also, pay attention to peak values for current spikes and the like. Average current is one thing, peak current and power handling capabilities are another.
"Do I only need to match package/voltage/current?" This is what designers do when looking for second source parts. In most cases, you will be good to go.
"Are Schottky and fast recovery the same?" Usually, fast recovery is a label for p-intrinsic-n diodes, i.e. Si diodes, that are trimmed towards blocking fast upon being reverse biased. Schottky diodes are, really, always as fast as can be and should not be replaced with slower Si diodes, even when they're labeled "fast". The general rule is: Don't replace Schottky with Si diodes.
"Can I sub an ultra-fast for a fast recovery?" Using an ultra-fast Si diode in place of a fast Si diode should work as far as switching losses go, but the faster switching action might cause worse electro-magnetic emissions.
"And... at the moment I need a Schottky that is in the TO220 package. Can I use two axial leads and just wire the cathodes together?" Diodes in three-pin TO220 packages are really just two diodes. They are, however, very similar. When connected in parallel, they will share the load really well. Different packages are also often a hint towards different peak current/power handling capabilities. And, of course, thermal properties will be different. A TO220 has its own little heat sink even when mounted in free air; axial diodes don't have this nice feature.