The Allied Navies in Norway

A large number of Naval units operated around Norway during the fighting there. Below I have listed the ones I have seen mentioned.

Aircraft Carriers

Furious    
Glorious   Sunk on 8.June
Ark Royal    

Battleships and Battlecruisers

Warspite   Second battle of Narvik on 13.April, bombarding Narvik on several occations.
Renown   Operation Wilfred from 5.April. Action agains Scharnhorst and Gneisenau on 9.April
Resolution    
Repulse Home Fleet  
Rodney Home Fleet  
Valiant Home Fleet  

Cruisers

Manchester    
Birmingham    
Devonshire    
York    
Galatea    
Arethusa    
Sheffield Home Fleet  
Penelope Home Fleet  
Glasgow    
Southampton   Assisted the Assault on Narvik on 28.May
Effingham   Grounded 17.May while transporting troops. Not salvageable.
Aurora    
Enterprise    
Suffolk   Seriously damaged by German air on 17.April after bombarding Sola airfield.
     
Cairo   AA-cruiser
Curlew   AA-cruiser
Carlisle   AA-cruiser, Arrived to late to stop German landing at Hemnesberget 10.May.
Curacoa   AA-cruiser
Calcutta   AA-cruiser
Coventry   AA-cruiser
     
Montcalm   French
Emile Bertin   French

Destroyers
A large number of destroyers participated in operations around Norway. Here I have listed the ones I have found mentioned.

Acasta Sunk 8.June in action against Scharnhorst and Gneisenau. Did however manage to damage Scharnhorst seriously.
Ardent Sunk 8.June in action against Scharnhorst and Gneisenau
Bedouin Second battle of Narvik 13.April.
Cossack Second battle of Narvik 13.April.
Eskimo Second battle of Narvik 13.April.
Punjabi Second battle of Narvik 13.April.
Hero Second battle of Narvik 13.April.
Icarus Second battle of Narvik 13.April.
Kimberley Second battle of Narvik 13.April.
Forester Second battle of Narvik 13.April.
Foxhound Second battle of Narvik 13.April.
Hardy 2nd Flotilla First battle of Narvik 10.April. Flotilla leader. sunk 10.April in Ofotfjord
Hunter 2nd Flotilla First battle of Narvik 10.April. Sunk 10.April in Ofotfjord
Hotspur 2nd Flotilla First battle of Narvik 10.April. 
Havock 2nd Flotilla First battle of Narvik 10.April. 
Hostile 2nd Flotilla First battle of Narvik 10.April. 
Gurkha Sunk by German aircrafts 9.April
Glowworm   Sunk in action with Hipper and two destroyers 8.April
Wolwerine    
Zulu   Arrived to late to stop German landing at Hemnesberget.

Other

Auckland sloop  
Bittern sloop  
Black Swan sloop  
Stork sloop  
Fleetwood escort vessel  
Vindictive repair ship  
Orzel submarine  Polish; sank German transport Rio de Janeiro on 8.April
Triad submarine sank German transport Jonia on 9.April
Trident submarine damaged German transport Kreta on 9.April
Triton submarine sunk German transports Friedenau and Wigbert around 10.April
Sunfish submarine sank German transport Antares on 10.April

List of Naval units that participated in Naval actions around Norway

Action at 8 and 9.April
A confused search for the German naval units that had been reported in the North Sea resulted in only two surface actions.

Glowworm Destroyer Sunk 8.April in action with Hipper and two destroyers. Sank when it rammed Hipper.
Renown Battle Cruiser Fought with Scharnhorst and Gneisenau at 04:00 on 9.April.
Orzel Submarine Polish sub. Sank the transport Rio de Janeiro on 8.April

First battle of Narvik 10.April
The Commander of the 2nd Destroyer Flotilla decides to try to take care of the German destroyers in Narvik. Unfortunately they are twice as many as reported, and the British are lucky to get an even score. Two destroyers lost on both sides.

Hardy Destroyer Flotilla leader, sunk
Hunter Destroyer sunk
Hotspur Destroyer  
Havock Destroyer  
Hostile Destroyer  

Second battle of Narvik 13.April
A bigger naval force enters Ofotfjord to take care of the remaining German destroyers. Low on ammunition the German destroyers don't put up much of a fight, and most are beached before they can be sunk.

Warspite Battleship  
Bedouin Destroyer  
Cossack Destroyer  
Eskimo Destroyer  
Punjabi Destroyer  
Hero Destroyer  
Icarus Destroyer  
Kimberley Destroyer  
Forester Destroyer  
Foxhound Destroyer  

Action 8.June
A German action against the British base at Harstad coincides with the Allied evacuation of Norway.
Scharnhorst, Gneisenau, Hipper and a few destroyers entered an area with a number of troop transport convoys, carriers and other high value targets. 

Glorious Carrier Sunk 8.June in action against Scharnhorst and Gneisenau
Acasta Destroyer Sunk 8.June in action against Scharnhorst and Gneisenau
Ardent Destroyer Sunk 8.June in action against Scharnhorst and Gneisenau