Using GDAL >= 1.10.0 compiled with SQLite and SpatiaLite:
ogr2ogr world_shifted.shp world.shp -dialect sqlite -sql "SELECT ShiftCoords(geometry,180,0) FROM world"
ogr2ogr -s_srs EPSG:4326 -t_srs "+proj=longlat +ellps=WGS84 +pm=-180 +datum=WGS84 +no_defs" world_shifted.shp world.shp
Both commands produce a longitude offset of 180°, i.e. a prime meridian of -180° is considered. In fact:
>ogrinfo world_shifted.shp world_shifted | grep Extent
Extent: (0.000000, -90.000000) - (360.000000, 83.623596)
The difference between the two commands is that with a longitude offset (2nd try) data are simply reprojected using -180° as prime meridian, while shifting the coordinates geometries (1st try) are altered, even if the result is apparently the same.
If there are parts in 0-180 that should not move, it's possible to adapt this working solution: https://gis.stackexchange.com/a/73164/22405
Clip the two parts:
ogr2ogr world_part1.shp world.shp -clipsrc -180 -90 0 90
ogr2ogr world_part2.shp world.shp -clipsrc 0 -90 180 90
Shift only the first part:
ogr2ogr world_part1_shifted.shp world_part1.shp -dialect sqlite -sql "SELECT ShiftCoords(geometry,360,0), CNTRY_NAME FROM world_part1"
Then, merge the second part and the first shifted:
ogr2ogr world_0_360_raw.shp world_part2.shp
ogr2ogr -update -append world_0_360_raw.shp world_part1_shifted.shp -nln world_0_360_raw
Finally, dissolve countries boundaries of
world_0_360.shp by country names. For instance:
ogr2ogr world_0_360.shp world_0_360_raw.shp -dialect sqlite -sql "SELECT ST_Union(Geometry), CNTRY_NAME FROM world_0_360_raw GROUP BY CNTRY_NAME"