There are a couple of other choices here.
Neither of which absolutely require that you provide a separate power supply (although now that you have already purchased one you might feel that you want to use it anyway).
First, you could just use a straight capacitive dropper direct from the incoming mains (but after any fuse).
Something like this for example:
https://www.nomad.ee/micros/transformerless/index.shtml
I have used this type of circuit many times in the past.
The warning about the danger of this type of circuit in that article is well made but is only really relevant if you are going to be messing about with it to drive something that you are likely to be touching. Enclosed within the case of a valve radio that is already swarming with interesting ways of electrocuting oneself it is no more dangerous than what you are already dealing with anyway.
(I must confess that I would reconfigure that circuit to put the main dropper part of the circuit - R1, R2, & C1 - on the live rather than the neutral side of the circuit but that's just me and my autistic tendencies.)
The advantage of this circuit is that it is very small and cheap. Note that, as noted in the article, C1 must be an X2 type - i.e. suitable for direct connection to the mains.
As a suggestion, for a 110V system for use with the LEDs that you have already purchased (and assuming 60Hz rather than the UK standard of 50Hz) I would consider adjusting the values that load on the page as defaults to the following values D1 = 9.6V, C1 = 1uF, R1 = 220 ohms. Note that this results in a voltage available to drive the LEDs of 8.9V. Your LEDs are likely to be screamingly bright and quite out of keeping with that required to look good inside a vintage radio if run at 12V. In fact, you may very well find that 5V (or even 3V) is a far more sensible voltage to drive them at in order to offer the best match to the original (typically rather dim) filament lamps. For 5V set D1 = 5.6V, C1 = 0.47uF, R1 = 220 ohms.
Second, the chances are very strong that the existing transformer within the radio is supplying low voltage AC (often 6.3V) to the filaments of the valves. You could fashion a simple circuit to tap off of this supply for some LEDs using just a single diode, a resistor, and a capacitor.
The benefit of both of the above circuits are (a) cost (but this is moot since you have already purchased a separate PSU) and (b) compact size (again because of the absent PSU so no need to accommodate the size of it within your existing equipment which can be inconvenient and / or difficult at time).
Option (i) is very useful for low current loads but R1 becomes big and hot as the required output current is increased. If you are only looking to drive three LEDs as mentioned in your post above then it should be fine.
Option (ii) involves tapping into the existing circuitry of the radio - something that you may not be comfortable with depending upon your level of experience with a soldering iron.
Personally, if it were my radio I would be going with option (ii) as the simplest and cheapest approach.
I appreciate that these suggestions fall squarely into the realm of "I wouldn't start from here" but I offer them up anyway just in case they should prove useful.
So, after all that, it maybe best to just stick with the PSU that you have already purchased as this avoids the high voltage of option (i) and the need to patch into the radio's existing circuitry of option (ii)! (Sorry!)
I would be cautious about trying to use the potentiometers that you have purchased as dimming / dropper resistors as such items are not typically rated for the power dissipation needed in this context.
What value are they?
What is their power ratings? This is not a figure that is typically easily available for potentiometers so should act as a bit of a red (or at least an amber) flag if not.
