There's a few exercises, and fitness regimens, that can increase speed, yes, that's right, actually increase speed.
Let's start with hand/arm speed. Here's one of them -
Start a training regimen of clap push ups. Not clapping in front, clapping behind your back. Sometimes when we start someone on this, we'll have them put a pillow under their face. And encourage them to turn their face if they aren't going to complete the push up - and end up face planting. If you can't complete one, there's an easy way to build up to that.
Start with a push up. Now, do the push up with a blast and have your hands come up off the floor. Do it again and clap in front, again, blasting off the floor with the hands. You'll see how much time you have, which is more than plenty. Then - do the push up and clap the side of your thighs and catch yourself. To make it easier, start on your knuckles and land on your palms. This will give you and extra beat, as you're starting higher and landing lower.
After you've done the clap of the thighs, it's just a few more inches to clap behind your back. Once you get it, and again, it's a lot easier than it reads, work the hell out of it.
Think about how fast your hand moves when you accidentally touch a hot stove. That's how quick you can be with your hands right now, which is pretty quick when given the right stimulus. You can even get that natural reaction slightly quicker with a good regimen of speed drills.
Side note - I hadn't actually done a clap-behind in a few years. So I just tried a few. Despite a shoulder that I tweaked yesterday, it was fine, and I'm an old man. Give them a try. Work the hell out of them. You should see a difference in speed in just a few months.
There's several other exercises that compliment this, we'll get to them later if you're interested. And one of them is with weights. And it's a corker.