Modifying a slicer

You would need to create a new table with 2 columns, 1 for the label you wanted in the slicer and 1 for the actual type. "Emergency or priority" would have 2 rows, one for each of the actual types. You could use DAX to create this

Slicer Table =
SELECTCOLUMNS (
    {
        ( "Emergency", "Emergency" ),
        ( "Priority", "Priority" ),
        ( "Routine", "Routine" ),
        ( "Emergency or Priority", "Emergency" ),
        ( "Emergency or Priority", "Priority" )
    },
    "Label Type", [Value1],
    "Actual Type", [Value2]
)

Link the actual type column to your main table in a many-to-many single direction relationship, so that the slicer table filters the main table. Use the slicer table in your slicer visual.