I bought a transistor/diode/capacitor/regulator tester from Frankie, it does the high voltage nicely, gives up to 1400V to test with. Though with 1000V diodes I would regard a safe point for breakdown to be the voltage half of which is needed to avalanche it or where the leakage is over 100uA. So for a 1kV nominal diode you need to have at least 2kV test voltage.
Forward current is easy, a 12V car battery and a car headlamp, 55/65W, with both filaments in parallel, and check forward voltage of each diode in a quick test. Use a DVM on the 20V range and check with each one that Vf is below the rated max, probably between 0.6-1V. Quick and easy to do, 3 seconds per diode will be an easy way to check for each one.