I'm writing a metal song that has a riff variation in a different time signature. I'm trying to figure out what the most natural BPM change would be so its less notable that I'm changing time signatures.
Since the new time signature still has the quarter note (4) for a beat, you don't have to change the tempo for the riff if you want to maintain the tempo of the song.
Instead if the new time signature had a different value for a beat, then you simply divide the product of the tempo and the value of a beat of the original time signature with the time value of a beat of the new time signature so as to maintain the tempo of the whole song. Example, instead of 4/4 if you take 5/8 or 6/8 then the new tempo would be (120×4)/8 = 60bpm.
Also, if you want a new tempo so to change the pace for the riff compared with the whole song you can take whatever you want immaterial of the time signature.
If you want it so that the listener keeps tapping their foot in time - 'most natural BPM change', then do nothing. The beats will continue in whatever tempo they were before the change, only the emphasis of where the first beat would come will change.
Say you go from 4/4 to 5/4. Or 4/4 to 7/4, etc. The 4 stays at the same bpm., all that changes is how many of those bpm fit in each bar.
Any other change will not be smooth. Changing things so that, say, your 4/4 bar now lasts as long as the new bar in a different time sig. is fraught with problems, too - the whole song will sound like it's fallen apart at that point - which, to a degree, it would have.Even halving or doubling will make the listener aware that something has changed, which apparently is not in your remit.
Short answer, at 120 bpm nothing needs to change apart from 6/4 going to 4/4.