The only part of the United Kingdom to follow the trend was Ireland after the First World War, and it is now a modern, secular, tricolour, democratic Republic.
Northern Ireland would like a word. Plus I'm not sure a state where abortion is illegal on religious grounds could be fully called 'secular'. I do have great respect for the Irish though.
Britain is a very arcane and quite anarchranistic a country. It in many ways resembles Austria-Hubgary more than any modern country, in the sense that is a union of different peoples, or to put it bluntly, an English empire. It also reassembles in many ways a pre-democratic, pre-enlightenment empire with all 'ye olde' institutions such as the nobility and city guilds.
Have Britain, around the time of the First World War, follow the AHE, the Ottomans and the German Empire, along with Tsarist Russia, to the knackers yards, with all four countries becoming independent and sovereign nations (England doesn't have to become a republic, but the three others do.)
Can I make a guess and suggest you don't live in England?
Everything 'ye olde' about England, or rather GREAT BRITAIN (Scotland, Wales and N.I. are still with us afterall), is still in place BECAUSE IT WORKS.
Seriously. Britain ADAPTS. Yes we have A LOT of hangovers from the past centuries, but that is because despite these things we have (in my opinion) one of the most peaceful, progressive and stable nations in the world.
If it ain't broke, don't fix it. We fixed the Royals in 1648 (though with a few bumps here and there after). We fixed (or rather disposed of) the Empire following WW1/WW2, even then Home Rule was gathering speed before it fell apart.
Scotland doesn't need a Tricolour. Wales doesn't need a Tricolour. N.I. doesn't need a Tricolour. Even Cornwall doesn't need a Tricolour.
And, most certainly, England doesn't need a Tricolour.
We have problems, but we will work through them, for we are THE United Kingdom. We work through our problems TOGETHER. United we stand, and may this union remain for centuries to come.