The plugs are all different, but the actual function of wired remotes are remarkably alike (with the lone exception of Panasonic - we'll get to that in a minute). Regardless of the interface used to connect the camera and wired remote, they are all pretty much simple three wire switches with a ground, a wire for 'half-press/focus', and a wire for 'fire'. The Panasonic connector only uses two wires and resistors to communicate the two different functions. It's all covered on this DIY blog.
For your Olympus E-M10 Mark II, this one or this one seems to fit the bill. This one and this one include built-in intervalometers.
There have been a few remotes with built in intervalometers offered in the past that include detachable cables for pretty much every major camera type on the market. The same remote is offered by a plethora of sellers with one of a wide variety of connectors attached.
For a more in-depth answer please this one to Is there a good remote timer compatible with most Nikon and Canon (and Pentax and Sony) cameras?
Regarding the user experience, I've had the same third party wired release with built in intervalometer for at least five years and it still works fine. I've used a few of the simpler generic wired shutter release cords with just a two-stage button that slides to lock. I had one that the connector to the camera came apart after a few uses but the others worked fine (at least until I managed to lose most of them). I discuss this a bit more in this answer.
Ignore the complaints in reviews on sites such as amazon.com about having to remove the batteries from the ones with built-in timers because you can't turn them off. They are like a digital watch: as long as the alarm isn't constantly beeping the batteries will last for years. In the case of the remote, be sure it is not running/counting down and actively trying to trigger the shutter periodically when you store it. You do this by setting the length and interval to "0" and the number to "1", then hold the 'lock' button until the "L" appears in the lower left of the lcd display. Even with no batteries the manual shutter button is still usable with most wired remotes with built-in intervalometers, you just can't use the timer part of it.