Will authorities in EU countries recognise the documents – or could I still be told I need to buy cover from a local insurer while I am driving abroad? 

EU member states will all recognise the Green Card document. Provided you have these documents, you will not need to purchase additional insurance from a local insurer.   

The Green Card system has a standardised format that has been agreed by all EU member states (including the UK) and is currently used for travel outside the EU to other Green Card member countries (you can find a full list of Green Card member countries here.)