I have an entry fire door which closes with a double chain closer.

The door closes very slowly and does not close completely. I have also noticed an intermittent "clunk" from the interior of the door about 10-20 seconds after the door has been closed. I'm guessing this could be the chain bunching up as the door closes, but would be grateful for any advice; particularly whether I should be tightening or loosening the adjusting screw on the frame plate on the right.

Fire doors should have approved self closing devices to BS EN 1154: 1997 and as far as I am aware all chain types do not conform, consequently you need to change it.

Some chain closers do conform to BS EN1154 and have controls for adjusting the closing speed and the latching action.  The self-closing device in the picture looks similar to the Perkomatic by Samuel Heath.

Look for a manufacturers mark on the self-closing device and contact them for the adjustment instructions.

Neil the only similar type of door closer that I am aware of is the perko powermatic, which dosent use chains, which conforms to BS EN 1154:1997, could you PM me, detail of others that I could research.

Did you ever find out about these chain closers the reason I ask is I don't want the standard top of door closer in a residential setting for adults with complex needs due to the risk of them using this as a point for suspending themselves for suicide any advice appreciated  

Samual Heath at  https://concealeddoorclosers.com/powermatic-r100-concealed-door-closer/

New Star at  https://www.new-star.co.uk/products/agb-eclipse-self-concealed-hinge-with-door-closer?_pos=1&_psq=self-closer&_ss=e&_v=1.0

Contact them and the door manufacturer to check suitability for your type of door, fire rating and end use.