1. When are your flights? LH and UA are both moving to T2, UA in June and LH in October.
2. Yes UA will make the change for free to fix an illegal connection.
I'm surprised there's nothing else available from MUC-LHR. LH runs quite a few flights per day. Alternative routings would include MUC-FRA-LHR or MUC-ZRH-LHR, plus connections through VIE (OS) and BRU (SN).