Could it be something from the Finno-Ugric family?

Definitely not Finnic-Ugric, but a language that’s heavily influenced by Spanish.
It may be spoken in Central America or on islands in the Pacific Ocean: Guam > CHamorro or the Philippines > Tagalog.

I’d say American of some sort – it’s not Tagalog or Chamorro

It’s not Tagalog or Chamorro, but is a Spanish-influenced Malay language, si U’d say one of the other Philipine languages– Ilokano or Cebuano, for example.

d.m.f.

It may be a minor Philippine language, but it definitely is not Ilokano and I don’t think it’s Cebuano or any of the Visayan dialects.

It seems nobody is going to get this one – the answer is Palauan, an Austronesian language family spoken in Palau, Guam and the Northern Mariana Islands.

The recording comes from the GRN.

Interestingly, the word for “Jesus” in this language is [jeʃuʃ] or [jesus]. Who converted them to Christianity? The Germans, who had Palau for 15 years?

Lev, I didn’t hear the “sch” sound at all. I heard yésus. I’m not very good with IPA, but I know that the man in the recording was not using the sh sound.