I can't speak for using a pick with bass, as I switched to finger style as soon as I picked one up. But when I used to play guitar - what helped me most outside of having a proper grip on the pick was practicing my weak points a little longer than I practiced anything else. For me it took me a long time (because I never got instruction) to realize my faults and the bad habits I'd developed playing.. when picking it's important to remember that when you're using alternating strokes to follow that through even when switching strings. Say your hitting a down stroke on the E string (random fret) and your next note is played on the A string, ideally you want to be able to comfortably hit that A string with an upstroke.. same for if you start on the E with an upstroke, you want to hit the A with a downstroke. Of course double down up, and other techniques work. But I developed more speed and consistency when I grasped this concept of alternate picking that I didn't understand for many years.
Less wordy examples - 5th fret E string down stroke, 3rd fret A string upstroke.. or 5th fret E string upstroke, 3rd fret A string down stroke. Paying attention and practicing changing strings with this method in your scales/riffs at a slow pace will ultimately be what makes you able to play faster.
Taking bass more seriously - I will say - Practice slowly before you try to practice at full speed. Even when I play an awesome new riff that sounded flawless, I find I have to go back and iron out the technique slowly and develop muscle memory before being able to consistently repeat what seemed effortless the first time. I'll wing it when doing improv with some drums+guitars but when I find keepers I take them home, slow them down, and make sure I can play them clearly at a slow and steady pace before trying to play it all full speed. I also look at it this way - when you don't take the time to slow down - it's harder to spot what bad habits might be taking place.
One last thought for when I would play fast with a pick on guitar - It changed the tone of my guitars somewhat, but, I found it easier to play faster when I turned my pick so that the tip rolled over the strings rather than catch and pluck them. It can be a drastic change on your tone, but, really made it easier to play fast. Dynamically it acts as a weaker attack on the string, but if you don't apply pressure the pick will just roll back and forth over the strings as you play. You may have already figured this out by now.. but you never know.
Not sure if I've been any help, but the very few times I've used a pick on bass I still feel the improvements I made on guitar when I started really looking at the flaws of my right hand technique.