Heart rate variability (HRV) is the most practical window into vagal tone. Many smartwatches, rings and chest straps track it (chest straps are generally the most accurate, and readings vary between devices), and a HeartMath sensor shows it live while you practise.
A few rules make the numbers useful. Measure at the same time each day, ideally first thing in the morning. Watch the trend over weeks, not the number on any one day, because a bad night's sleep or a big dinner will move it. And compare yourself to yourself. There's no universal good score: HRV varies a lot with age, fitness and the device you're using.
How long does it take? You'll feel the in-the-moment shift from slow breathing within a few minutes. The baseline change, the vagal tone itself, builds over weeks of consistent practice. Studies of slow-breathing practice usually run for somewhere between four and twelve weeks, so give it at least a month before you judge it.