First of all to kill those CL's the UK needs CA's and BC's , the UK's established policy since the first Churchill Reign in the Admiralty in 1914, was to bring overwhelming firepower to combat the deficiencies in crew, training, firepower etc vis-a-vis the German Navy. A CL vs CL battle was not something the RN enjoyed as it was not without the risks.
Like the force that went after Graf Spee? They were hardly overwhelming. There are multiple occasions during WW2 of Commonwealth forces fighting at even (or worse) odds as required. I agree they very much focussed on overwhelming force in WW1, but WW2 was very different (in practice at least, I can't speak for doctrine, just for whether it was followed or not).
Edit: And it's also worth noting that if Germany did build a lot of CLs, then the UK would likely have built more CAs.
For eg: those German Merchant marine raiders were all hunted down using CL and CA which were much bigger and had 3X the firepower of those smaller and less armed ships.
There were 11 merchant raiders, their fates were:
Orion - sunk by the RAF
Atlantis - sunk by CA (Devonshire)
Widder - converted into a repair ship in 1944 and survived the war
Thor - blown up by an explosion (likely accidental) in a Japanese port
Pinguin - sunk by CA (Devonshire)
Stier - disabled by US Liberty Ship (Stephen Hopkins - sunk by Stier during the action) and scuttled by crew
Komet - sunk by MTB 236 in the English Channel
Michel - sunk by SS (USS Tarpon)
Kormoran - sunk by CL (HMAS Sydney, which was also sunk by the Kormoran)
Coronel - damaged by bombing while trying to break out to Atlantic, returned to Germany and converted to Fighter Direction Ship
Schiff 5 - never became operational due to bomb damage.
From this, it looks more like that they were stopped by whatever was to hand (although Stephen Hopkins' work clearly wasn't what they were hoping for!) That said, the cruisers were clearly on the lookout for surface raiders (that was, by no small means, what they were designed to do) and would have done a lot of the work, but not because they were trying to apply 'overwhelming firepower' (again, see Graf Spee) but because they were the ships tasked with (and designed for) that role.
The Japanese entry was precipitated by another blunder of Churchill and the UK's high command, the story is fascinating and wiki has a small article for introductory purpose-
https://en.wikipedia.org/wiki/SS_Automedon
I'm not sure the relevance of this - I'm assuming a WW2 that breaks out along historical lines, and my point was that if this happens, German surface ships mean the UK keeps more ships in the home fleet, and so has less to fight Italy and Japan with. That said, I thought the US oil embargo had a little bit to do with their entry into the war as well, although happy to hear arguments to the contrary.
In that circumstance (Germany, Italy and Japan allied in a war against the US and UK, with the USSR and Germany going at it as well), if the best Germany has is a CL, the size of the Home Fleet need not be anywhere near the level that was maintained historically. A couple of Revenge class BBs with a few CLs and destroyers would cause all sorts of trouble for a CL-based force, and there's no need to have BBs on convoy duty (and, as air power over the Atlantic intensifies, CLs would become pretty much obsolete for convoy raiding in any event, as it becomes too easy to identify and vector forces in to sink them). If those ships aren't with the Home Fleet, then they're causing more pain for Italy and Japan.
Again, the UK sending some more ships to Singapore will end up in even more losses for the RN, the Japanese had sent their "Sledge Hammer" to take Singapore and Malaya and only an all out effort like the US did at Midway will help, this means all the available carriers and bulk of the Fleet which was not possible since the UK considered the Home Fleet and Med Fleet more important (rightly). Adm. Sommerville rightly ran away from the Japanese despite having been told to do the contrary as he was a Wiser commander than those sitting in London.
You'd only need a couple of CVs to make a substantial difference at Singapore - as far as I can recall, none of the Japanese CVs were near the area, and while I can't recall the exact forces available, I think the core was a couple of BBs*. The air strikes on Prince of Wales and Repulse were carried out by relatively small numbers of bombers (and IIRC - but could be wrong - got lucky with both ships having technical issues with their AA). Put 30 sea hurricanes in the air as a CAP, and the Japanese landing forces on Malaya would be in a tough spot, and the Battle of the Java Sea (the largest ship Japan had there was a CA, albeit it had two very good ones) could end very differently.
* Your memory of this seems to recall a larger fleet - I had a bit of a look around and couldn't find anything, but happy to be corrected if I'm off the mark

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As for Sommerville, you're quite right in that he avoided the main Japanese force, as he was outnumbered in almost every class, but my point is that if Germany has nothing to encourage Britain to maintain a strong fleet at home, then this need not be the case. Another thought is that it was only Germany's surface vessels that gave it any chance of mounting a successful invasion - if an invasion isn't a credible threat, then the RAF can also send more and better aircraft to the Far East sooner, and there's less need to maintain as large a land force in the UK as well (although I doubt this would have been overly large in late '41 as well, although don't know).