Thanks to some help from Tuan I have managed to track down the problem (it wasn't a problem with Membership Pro).
There were two separate issues as it turned out.
Username not populated from Membership Pro / Joomla User - fixed by enabling the plugin called "AcyMailing : (auto)Subscribe during Joomla registration". Strangely I have two other websites running quite happily without this enabled!
Confirmation email not sent by Acymailing on membership subscription activation - fixed in Acymailing configuration screen, subscriptions tab by specifying a list against "Auto-subscribe on site account creation". The confirmation email will only get sent if a list is selected here. The user won't actually get subscribed to that list, only to the lists specified in Membership Pro against the subscription plan.
