There is a significant chance that LH will cancel your itinerary well before T-24, because it hasn't been ticketed.

I don't think it's possible to do what LH is asking. They're basically asking UA to reticket this as a business class ticket despite UA not having received payment for a business class ticket. The reason that UA is telling you that they can ( / might be able to) do this at T-24 is that a lot of the ticket validation restrictions drop off at that point in favor of getting the passenger where they need to go. They're basically suggesting that if you wait, they'll treat you like they would treat a displaced passenger, and put you onto that flight even though your ticket doesn't qualify you for it.

UA allows you to upgrade UA-operated segments on non-UA tickets, but they do that without involving the ticketing airline -- they just update their own system and note the payment. As far as the ticketing airline is concerned, the reservation is untouched. For LH to send you to UA suggests that they don't have the ability to do this, and, as such, suggests a significant likelihood that this isn't going to work out in your favor.

I would suggest following up on the LH board to see if other people have been successful with this. If you want to move this post, you can use the little red triangle to alert a moderator and ask them to move it for you.