Loading each winding reduces the measured voltage due to internal winding resistance. Waiting say 30 mins to an hour allows the copper winding to increase in temperature to a level more likely experienced during normal operation, and that increases the winding resistance a bit, which also would lower the measured voltage. The 5V winding would typically be good for 2A, and possibly 3A, as they were the common rectifier heater current levels.
The physical size of the core will indicate what VA rating the transformer may be good for - that requires comparisons with known transformers. Measuring the DC resistances of the primary and secondary HT windings can also be a reasonable comparison with other known transformers.
Loading the 6.3V and the 380-0-380V windings requires a bit more technical effort to derive a reasonable load resistance to apply, especially for the 380-0-380V windings as they would be rectified and filtered in a normal amp. If you aspire to a KT88 PP amp then you may want to use a 4A loading on the 6.3V to confirm ok operation. Although you could put a resistor across each HT half secondary (380-0V) winding it may be better to set up a temporary ss rectifier capacitor filter and load the DC output - say to a typical idle current requirement like 100mA. A lot of effort to make, but it can avoid making up an amp and then realising the PT doesn't actually suit.