It’s so frustrating and implausible how it’s implied that, before Kat, NO ONE had a clue about how the professor and Lukes’ disappearance.
I have no problem believing that the police failed to investigate the disappearance. From the prequel games to the LMJ time, we see that Scotland Yard is incompetent at best and corrupt at worst. You have some good officers like Inspector Grosky, Inspector Chelmey and Barton (who has become Commissioner by the time of Mystery Room), but even they depended on the professor for help with their cases.
Then you have the bad ones... Detective Bloom (Targent mole), Justin Lawson (helped a serial killer, brainwashed Alfendi) and that Inspector Britannias (I’ve probably spelt his name wrong) who tried to frame Kat.
Scotland Yard was still corrupt during the LMJ time period, and it’s very likely that they sabotaged/forged any evidence linked to Layton and Luke’s disappearance. My main bet is that Justin Lawson was concerned that if Professor Layton returned he would uncover his crimes. Or maybe Justin just wanted to distract Alfendi with the hopeless search for his father.
It’s never confirmed in LMJ, but we can assume that Emmy, Flora, Randall and ESPECIALLY Desmond were investigating behind the scenes. Maybe they would have found Layton and Luke sooner … if only the police had been more helpful.