It does seem O2 is having mast congestion issues at the moment as they upgrade their masts.
Network Future http://www.o2.co.uk/connectivity/network-coverage
If sim swaps etc don't work then if your not locked into a contract or on Refresh you could pay up your handset plan and move networks.
You could try out alternative network Payg SIM cards to ensure you get good signal where you live, commute and work and perhaps migrate your number to another network via a pac code. ->
Unlock your phone for free for contract customers or £15 for Payg customers - T&C apply via this form ->
https://www.o2.co.uk/apps/help/help?qid=1&q1=2&route=unlocking&case=Handset%20Unlocking%20Form to try alternative SIM cards.
I'm quite pleased that my write up on sim swaps is to your liking @MI5 😄
Can I ask if you have TuGo enabled and the messages are coming through that on WiFi. It would seem to be the only scenario that you describe.
Track usage, upgrade and get support with the My O2 appDownload My O2