I was in a very similar situation. Mine was BHX-MUC-KRK
first flight left 45 minute late and missed connection. First email was exactly the same but I challenged it and within a week they decided to pay me €800 (400 per pax).

I actually came here for advice on who to contact regarding payment. Been a couple months now, they were very very quick in asking for bank details and now I'm ghosted. I've emailed back and asked for update via website but nothing.

I'll leave it a bit longer but am I better referring this to the UK ombudsman or German?

Thanks