So there was an original planned musical with music from Leonard Bernstein that was first staged in 1950 but had the songs cut down to 5 to play more like a straight drama. Probably what then happened is both Peter Pan the Disney movie probably had been in development long before that (it was actually in Disney development hell for what looks like nearly 15 years). It looks like since the 1950 version was a drama then another producer could secure the American musical rights and thus both the animated movie and the 1953 musical were developed independently but on parallel timelines.