I'm working on teaching both of my horses to get their flying lead changes. Depending on the horses ability to learn, the time to get it varies. I've been working with my horses on getting them responsive to the leg, working on the right leads and asking for counter cantering, switching back and forth. For most horses, flying lead changes come naturally, however some horses I've seen aren't coordinated enough to do them.
Practice changing leads by going down to a trot and eventually dropping the amount of strides in between the lead changes. Don't rush the lead changes though, it's more important that the horse is balanced through the changes. You can also ride on a figure 8 and place a pole where the 2 circles meet and changing directions/leads over the pole. Just ask for a lead swap when the horse crosses over it.
You can't rush through a flying lead change, it will teach your horse that it's acceptable to speed up every time you do a flying. I've rode a horse that became really sucked back when asked for a flying because her owner practiced flyings too often.
For lower level showing it's okay to go without flying lead changes. But when you get to a more advanced level, it's imperative that your horse knows how to do them. For example, if you were a show jumper on a 2'6 course with a horse that didn't have flying lead changes, at that level you would most likely wouldn't place well. It takes a valuable seconds to pull your horse down to the trot and then go back up to the canter.
In reining patterns, the horses are required to do flying lead changes. If you're in a western pleasure class and your horse blows your lead, it only takes a few seconds to switch it through a flying where it'll take more time to correct it through a simple lead change.
My coach has a horse whose flying lead changes feel just like a step, you can hardly even tell the horse switched his lead. For other horses, not so much.
It's also really important that you don't rush the learning process. The horse will learn when it does, and trying to force it into a change or trying to force it do the change before it's ready could really hurt the horse. Also, make sure your horse swaps everything because some horses only swap the front lead.
I would advise you to get a trainer's help, even just taking a couple of lessons to work on flying lead changes would really help and avoid injury.