Thought I'd follow-up on this as I've finally tracked the problem down and thought it worthy of posting about in case anyone finds themselves in this rather odd situation...
Whilst I was having problems with Roundcube via Chrome I was also having other issues with TinyCamMonitor (an IP camera app) on my Android phone and access to a locally-hosted Zoneminder (CCTV motion detection) installation. Looking back, the symptoms were similar insofar that there was occasional lack of responsiveness, but only when on the LAN and not accessing remotely. What I didn't do was spot the connection between all these symptoms and suspect a common cause. I also didn't put enough significance on the fact that I never used to get these problems - indeed all these services have been running for many years without issue - so what had changed?
Well, at the end of the last year I received an 'upgraded' VDSL modem/router from my broadband provider. I've put 'upgraded' in quotes as whilst it might've provided better WiFi performance it turns out that whilst it supports NAT Loopback it evidently doesn't cope too well when stretched with a high number of connections being tromboned through it. Swapping back to my old router has solved all my problems.
To help with keywords for web searches, the new modem/router is a Plusnet Hub One which is actually a rebadged BT Home Hub 5 and sure enough I've also tried a 'real' BT Home Hub 5 with exactly the same issue. NAT loopback works, but not when pushed. My old (they stopped making them 5 years ago I think?) Thomson TG582n router (with seperate BT Openreach VDSL modem) works fine.