How can I cut a metal pipe while preserving the wires inside?
You have to pull the wire out. Period.

What you're looking at here is the conduit wiring method. It's actually a very excellent way to run wiring, but I gather it's completely alien and unfamiliar to you.  

The gist is that you build the pipe route, then you run the wires through it, in that order.  The wires can be any color and size you want. 

Now, Code requires every junction box cover remain accessible without tools, forever.  It can't be covered with drywall, wallpaper, carpet, a screwed down plywood panel, not anything like that.   So Wherever the other end of that conduit is, you should be able to access it. That is where you pull the wires back from. Expect them to be individual wires, not the cable you are familar with.  

You also see a transition from EMT metal conduit to some sort of flexible conduit system such as AC.  I don't know what's going on with the tape, but I assume/hope there is an appropriate coupler there.   Also, I don't see what's happening with the ground wire.  Perhaps this flexible conduit is not allowed to carry ground (EMT is), and that was their fix.  The right solution is to add a ground wire to the inside of that conduit the entire run.  

The answer is to reroute that EMT and flexible conduit so it uses a different path.  Don't be bashful about getting more EMT or flexible conduit to extend.  They sell pre-bent sections and they also have conduit benders.

Do not exit the conduit with bare wires.  Honestly I think the metal conduit is a good idea due to the exposure to damage there (from the washer being installed and removed).  You are generally allowed to exit conduit with Romex cable, but not here, because it would be subject to damage.  However pulling stiff, balky cable through conduit is a major pain, and is often not practical for a DIYer. 

So find the other end of this conduit. Bind a rope to your wires at the end. Then pull the wires to pull the rope into the conduit. Then do your work, then use the rope to pull wires back in.