The voltage rating matter won't as long as it's sufficiently high. Whether it's multi-layer won't matter either. What will matter is the material type of capacitor, especially in something like a radio.
Something as small as 10pF in a radio, is probably in the signal path so it's going to have to be accurate, stable with temperature and applied voltage, with good frequency characteristics. Use something at least as good as a C0G/NP0 ceramic for something like this. If the caps you already have are C0G ceramic then you probably can use them. I don't know of any ceramic caps more stable than C0G. The next step up are things like film caps or mica caps which would also likely work.
The other types of ceramics aren't good enough for something like this. They have names that look like X7R, X5R, Y5V They aren't stable enough for signal-path stuff and are only sufficient for power decoupling.
I should note that 10pF is so small that it probably is a C0G/NP0 cap unless you explicitly know that they are not. That capacitance is so small that it's pretty much useless for any application that might need it unless it's C0G.