This depends on the capabilities of your PWB manufacturer.
Most can manufacture down to 0.1 mm (4 mil) space and trace dimensions with no impact on cost. Really low-cost manufacturers might require larger dimensions. If you pay more (or you're already paying more for finer features elsewhere in your design) many manufacturers can do at least slightly smaller.
Also, if you're using heavy copper (more than 1 oz or 35 um thickness) you might need to use larger dimensions.
For a definitive answer, contact your manufacturer.
In your question you said you're using Altium and in comments you said, "My CAD program doesn't seem to distinguish between vias and TH when forming connections to internal planes." This isn't correct.
In the rule editor, for power plane connections, if you choose the "advanced" rule type you can make separate rules for pad and via connections:
If you're using a very old version of Altium that doesn't offer the "advanced" rule type, you can make two rules and use queries to apply one to pads and one to vias. If I recall correctly, there are "IsPad" and "IsVia" predicates available to specify which rule applies to which type of hole.