Bare lcd of D821 is compatible with D820.
But their lcd screen supporting frames are different.
The lcd, digitizer and frame are glued together. It is very hard to replace only a bare Lcd unless you are professional. You'd better to replace a complete lcd with digitizer and frame, which costs just a little more.
Share with u the [http://www.witrigs.com/oem-lcd-screen-with-digitizer-assembly-and-frame-for-lg-nexus-5-d820|LG Nexus 5 D820 display assembly] and repair guide.