CAA PLD aims to get new licences out within 10 days of receiving an application. Assuming, that it, there are no errors in your application.
You can fly (but not take passengers) whilst waiting for your licence to arrive if you do so under the supervision of a FI.
But if you turn up at your local Club announcing that you have just completed your training elsewhere and are waiting for your licence to arrive, it wouldn't surprise me at all if they told you to go away and come back when your licence has arrived! Or alternatively, they may restrict you to flying with FIs until your licence has arrived - so that you can learn UK rules and procedures safely.