The danger when it comes to supplying this small amount of proprietary ammunition is not that the logistic system will break down entire, but that those specific, very important, supplies of ammunition will not make it to that specific formation, which I would argue is a very real possibility. In such a situation the effectiveness of that formation is significantly degraded.
If an entire Army Group, Army, Corps or even Division was going to a new round it might actually be easier as certain segment of the logistics system can then be dedicated to the new round. Though this is also the type of situation where broader breakdowns are possible.
Usually not is frontline service if they could help it. The British had the .303 in the front and the .30-06 and 6.5 Arisaka in rear echelons. The French had the 8 mm Lebel in the Front Line and the 11 mm Gras in some rear echelon and gendarme roles. The Germans had the 7.92 Mauser in the front and maybe some 11 mm Mauser in the rear. Austria Hungary and Russia are maybe the best examples of your point but my understanding is that they had so many calibres in front line service more due to desperation than choice, and I don't think anyone thought it improved their performance.
To sum up, I have to disagree with you here. IMO this would represent a non-negligible, and more importantly, unnecessary burden on the logistical system. Especially when the same company can make the same rifle in your standard calibre. I can't say for sure that Churchill would not do it, though I don't think he will. I know I wouldn't do it were I in his position. Even if I were the brigade or Divisional commander I think I would tell the London Scottish that they can draw SMLE's when they are under my command, and send the .280's back to help train the other battalions in their regiment. But that is me, and I am thankfully not Churchill (for one thing I have a hard time picturing him typing on a computer)