We don't know a ton about the final design for system-to-system jumps yet, but the original concept is that you have to actively fly your ship through a somewhat twisty "jump point" wormhole thing and come out the other side. That would be long enough to cover the loading time, but it's all active gameplay instead of loading screens.
It also won't need to load the whole system when you jump in, only the "low poly" versions of planets and moons and whatever large scale spacescaping objects (gas clouds, etc.) are visible from where you come out. The more detailed versions of planets and moons only need to stream in once you pick where you're going within the system spool up your quantum drive to get there.
If you don't want a "quantum drive" scene going from one place to another, you're also free to make a seamless flight using regular engines! Port Olisar to Daymar takes a bit under 10 hours.
https://www.youtube.com/watch?v=C_iXlgxsEVc (timelapse of that trip condensed to 15 minutes)
And if you're in something bigger than a single seat fighter, you can get up and do other stuff around your ship while it's traveling, but it's not a good idea in areas with pirates. Still, more options than a Mass Effect elevator.
Flying around a planet/moon all the way down to the surface, getting out of your ship and walking around, and going inside the various outposts happens completely seamlessly. I've only fired it up a couple times since beta 3.1 came out, but my first trip down to Yela was pretty mindblowing.