In the following snippet, two parts have a completely different time signature, yet remain synchronized.
The bar lines can no longer be printed at the Score level; to allow independent bar lines in each part, the Default_barline_engraver and Timing_translator are moved from the Score context to the Staff context.
If bar numbers are required, the Bar_number_engraver should also be moved, since it relies on properties set by the Timing_translator; a \with block can be used to add bar numbers to the relevant staff.