My Plan is to drive them via an UNO, can I run them on something smaller like a Nano?
You have use the word "drive" for different purposes here.
You cited power converters to supply regulated 5 V power to the WS2812B strips. That sounds reasonable if you use a multi-cell lithium battery pack - which will require a corresponding purpose-built charger. There is no necessary connection between the power supply used to supply the LED strips, and the Arduino, the Arduino is in no way whatsoever serving to "power" the LEDs.
It may however be that you use your well-regulated 5 V power supply to power the Arduino via a separate connection of the paired 5 V and ground wires directly from the power supply to the Arduino 5 V input terminals.
Then you use the term "drive" to generate the instruction waveform. This requires very little power and only connects to the first WS2812B in the strip. There should be a series resistor - about 470 Ohms - in series with the data pin of that first LED.
A UNO and a Nano are electrically virtually identical. Actually, a real UNO has a second processor chip - a 16U2 - as the USB interface while many or most fakes branded and sold as UNOs are not UNOs at all but a mutant of the earlier Duemilanove which used a deprecated USB interface chip and was electrically identical to the Nano apart from a couple of connections.
So the "clones" now fraudulently sold as "UNOs" and those sold - in this case not unreasonably - as Nanos now both use the more readily available and much cheaper (Chinese designed) CH340 USB interface chip and these versions are again, electrically the same apart from a couple of details in the power supply and pin 13 indicator circuits.
So board size is in this regard, completely irrelevant to their identical functionality - there is absolutely no difference whatsoever in "computing power" or electrical power usage. And for any serious project, the Nano is the preferred module.