Upgrade with printed e-SWU is valid on any international LH flight (booked under any flight number) on a standby basis. Sometimes they clear them at check in or in the lounges, sometimes at the gate, and sometimes (often) you don't clear. You have to stand by for each flight segment with its own printed e-SWU.

This certificate voucher is also valid on all international Lufthansa operated flights worldwide for standby upgrade.

For the UA flight you can upgrade at the gate using an e-SWU since it is booked on a codeshare. You could also use another upgrade certificate type at the gate for the UA flight since it is domestic. (I do not think you can use miles, though, at least officially.)

P.S. The seat is no different on most European flights in "business" except that the middle seat is left empty. So it is generally not worth the upgrade certificate for those.

Charles