The ticket stock is entirely irrelevant in this aspect. It does not matter if the ticket is issued by Swiss (724 stock I think) or United (016 stock).

What matters is if the fare class you purchased it upgradable as per the rules of the operating carrier. I.e. UA allows for upgrades from W fares with SWUs, LH does not allow intl. upgrades from W. LH/LX allow upgrades from V or Q class with miles, UA does not.

Now there is one big exception which kills all hope: codeshares. If you flight was booked under an LX flight number but is operated by UA, then this will slim your chances for an upgrade very substantially as there is no way to even request that upgrade until check in - read: until all upgrade requests by others have been cleared.

If you are booked on a codeshare, I would either a) bury my hopes and not do anything or b) change the flight number to the UA flight number.
I would not do the straightforward thing and wait until I can check in the earliest (say online 24 hours out) and then get UA instruments - as you can only use those - and pray for an upgrade.

Assuming that you did NOT make the mistake of booking a codeshare, then you can use US miles to upgrade IF you are booked in a B or Y class on UA.
Or you can also get a UA frequent flyer to upgrade you with miles when you are booked in M,H,B, or Y class. Or with an SWU when you are booked in W,V,Q,H,M,B, or Y.

Good luck!