I have a copy of the Jenkins song. FPB isn't borrowed, it's the same tune from start to finish.

The film dates from 1953 so Cash would have seen/heard it in Germany.

Vocals are by one Beverly Mahr (1953)

Lines from the Jenkins song that ended up in FPB include;

'I hear the train a comin', it's rollin round the bend'
'When I was just a baby my mama told me (Sue)'
'I'm stuck in (Crescent City)'
'When I hear that whistle blowing, I hang my head and cry'
'I see the rich folks eating in that fancy dining car'
'If I owned that lonesome whistle, if that railroad train was mine' (In fact almost the whole of the last verse is a steal.)

I would say the around 80% of FPB is a direct lift from Jenkins. What I find strange is that, for decades, FPB has had a writers credit solely of Cash.