Would it be more effective to do jogging or walking recovery between intervals/repeats in order to get faster?

That means you're running them too fast.

Generally jogging is the best approach if your recovery is > 60s. Under 60s doesn't really matter much. Or if you are doing very high intensity work, sometimes walking is appropriate because you need recovery to be longer and more complete. But even then it's usually some walking immediately after followed by easy jogging to lead into the next repeat.

Some say easy jogging between the sets allows you to use the "lactate buildup" as fuel.

Mile reps are pretty long for vo2 max work unless you are pretty fast, Jack Daniels recommends capping them at 5 minutes. The downside to very long intervals is exactly what you experience, even with 3 minute recovery jogs it's hard to actually fully recover.

For mile-5k paced reps of decent length I like to walk the first 20s or so of the recovery until I catch my breath, and jog the rest. For short intervals I just jog. See this article for some arguments on either side:

trailrunnermag.com

Why You Should Jog the Recovery in Interval Workouts

Reagan Colyer

A new study compares active and passive interval recoveries, but physiology isn’t the only factor to consider.

Link:www.trailrunnermag.com/training/workou...

But for some reason, the author argues for jogging recovery even though the study he cites says that standing/walking it leads to more time actually spent at v02max during the interval.

Sorry I just reread your post and noticed it said "mile repeats OR vo2 max intervals". If you mean mile repeats at threshold, I agree with the first poster, you are running them too fast. That's a pace you should be able to hold for probably around 10 miles in ideal conditions, if you are gassed after 1 mile then something is wrong.

Similarly with vo2 max intervals, if they are 400s with short recovery you should not be gassed after the first 2. But if they are 1ks or 1200s then yeah it's going to be tough, that's when I walk for 15-30s then jog the rest.

I just do walking/standing rest. It's about hitting the desired energy system over and over during the repetition, with enough recovery in-between to keep going.

I have never completely stopped between intervals. I always jog through to the start of the next one. That's the way I have been doing it for 30 years

You need to sprint hard in between the intervals.