Thanks for trying out the JMP mapping with the Brexit Results data!
As you have seen JMP includes some maps with the package, but only down to the county level for the UK.
To get down to a lower level (in this case the Area (city/town) level that is in the Brexit Results, you would need to bring in a custom shape map that contains the info to draw each of these Areas. Once you see the underlying tables needed (naming and shape files) you can create your own (a little time consuming) or import in and adapt an already created shape file (probably the best option here). Then put these into the folder your are mapping from in JMP and you should be ready to go. Here is a link to learn a little more: http://www.jmp.com/support/help/Examples_of_Creating_Maps.shtml
There is another way to view the data that might be helpful as well. If you add both a representative latitude and longitude columns to the datatable for each of the Areas, you can then plot these on a Graph Builder of Bubble Plot over the underlying built-in UK county shapes or any other geographic map. Not the same as having a break out map shape for each Area as you would with a custom mapping view, but might still give you the visual you want.
Shown above is a quick example on just a few rows of the Brexit Results data.
Hope this helps!