Hi Gerry,
I have dealed with same situation. Basically what we have done (since we have a gTLD .net) is that we have created sub-folders for each country and geo-targeted each sub-folder from GWT to the specific country.
Now, people say that having your own ccTLD (country codeTLD) can help you even more in rankings for local searches.
Another factor that you should consider, is moving the hosting of subfolder (if you are going for the first version) or ccTLD (if you use the second one) on a hosting from the targeted country.
P.S. if your client has a .co.uk domain (which is ccTLD) you can not geo-target a sub-folder from Google Webmasters Tools to another region, therefore I would advise you to buy their brand name in each country, check for a hosting, then start optimizing for the specific country.
I hope my answer helped,
Istvan