Not really - if you try, you could do 4 or 5. UK, Fiji, Tuvalu (12 chars) take maybe 2 seconds, then Australia and New Zealand (19 chars) may take 2/3 seconds more.
Of course, we must consider the time it takes to recall them. I got 6 seconds without trying to do it fast, and three while, and I'm no speed typist. I think 10 seconds may be too easy!
I feel like 30 seconds would be good. For these you either know them or you don't (many wouldn't recognize Fiji or Tuvalu) and a full minute allows certain people who don't know them to look it up. 30 seconds is good for people in case it doesn't come to them right away and a little bit of thinking is required.
How about adding the flags from Countries 197 to 205 as well? There's Cook Islands for sure. Would Niue's qualify, with the way extras are sprinkled into the Union Jack on their flag?
This might be a good candidate for a yellow box quiz. Most people have a pretty good idea of what countries have the Union Jack on them, but can you actually identify which country each flag belongs to? People can have a little fun trying to remember which one is Australia and which one is New Zealand!
I think this quiz would be more interesting without the flag images. - just name the 5 countries which you would need to do from memory. I seriously think people would forget the Union Jack itself.