Have a look at the total picture. It's good, but it's beginning to fall short of the government's highly ambitious target, it's not improving as it needs to - and single countries having a success is totally pointless.
Target = Offer everyone in Groups 1 - 4 a first vaccination by 15 Feb "the government's highly ambitious target"
Number estimates seem to vary but say total 15M. Reduce that by 20% refusals = 12M. Add a million for 'bias' = 13M
Data below from:
https://coronavirus.data.gov.uk/details/vaccinations
As at 29 Jan total first doses: 8,378,940 (in arms, not 'offered')
Daily = 365,656 (7-day average) (first doses, in arms, not 'offered')
Number of days till 15 Feb = 15 (say, could be 16 or even 17)
Reasonable estimate of next fortnight capability - 5,484,840
Reasonable estimate of numbers of first doses administered = 13.8M
The vaccination rate doesn't need to "improve" (though that'd be excellent) to actually overstep its target.
Edit: All those involved in procurement, distribution, storage (at the various levels across the nation), management and delivery (into arms) deserve massive plaudits. This is not a short race though; but by the end of spring the UK should be close to herd immunity (which is of course the overarching aim). Internal transmission will diminish and infections brought in from abroad will not gain a foothold and should be much easier to control, with minimal loss of life or serious illness.
Don't know what you mean by 'single countries' - is this Scotland, Wales and Northern Ireland? If so they seem to be reasonably on track to play their part in the United Kingdom's effort/programme.
If you're saying that the UK (a "single country") getting its act together (for once) "is totally pointless" then allow me to disagree. The UK efforts to help find/develop, initial manufacture, trial, manufacture at scale, an effective, inexpensive vaccine will save hundreds of thousands of lives across the world, not just in UK.
What do you suggest UK changes in its immediate future plans, or should have done differently with regard to vaccine development, procurement (selection), authorisation and roll out?