Evacuation from the entire country would only be a very last resort. However it is a given in your story, so we will work from there.
It's around 1895 and the British Isles have fallen to a communist uprising. Or sunk underwater. Or invaded by Germany or covered in ice or something. So the government, the Royal Family and a large part of the population has been successfully evacuated to the colonies.
Communist uprising: I find it unlikely that communism would spread to such an extreme in Britain, given the mentality at the time and the pride in the empire. If the mindset of the people were such that this was possible I don't think the British would have been as successful in building an empire. Also I find it unlikely that the rest of Europe would stand by and let an influential power like Britain fall to communism - doing so would embolden communists in their countries too.
- A second point on a communist uprising is that they may well attempt, once they have taken Britain, to take some of the colonies too - limiting our options of where to go.
Sunk underwater: Well this one gets marks for being a clean slate, no enemies to think about just a "Well you just can't pick Britain anymore." However I'm not sure I need to point out how impossible is it for a whole country to just sink....but as stories go, you don't need everything to be believable.
Invaded by Germany: Okay, so this could be possible (though Germany would strike closer to home in an initial attack, long supply routes through countries that may object to your war are expensive). However I don't think the British would leave/would be allowed to leave. You say you want a large part of the population to be evacuated but why would the Germans not just sink those ships? It is likely the British will be back for their country before too long.
Covered in ice: Like sinking this has the benefit of being a magic wand we can wave and say we have no choice, however it is hard to see how the rest of Europe wouldn't suffer a similar fate (thus changing the whole dynamic of world power and making a prediction of where to go mostly guesswork).
I'm going to assume we're taking the magic "Britain just isn't an option anymore"
The Royal Navy is still intact and still the largest in the world. The British Empire is still the largest Empire in the world. But what I'm wondering is where would the capital be? Singapore? Ottawa?, Bombay?, Cape Town? And how much of a world power would the British Empire be without Britain?
Here are the limitations as I see them:
Short journey: There are a lot of reasons for this but primarily:
- Disasters at sea aren't unheard of and transporting people, wealth and the monarchy across large distances increases this risk.
- Pirates, think of that wealth again, it is likely some pirates (or other European powers) would attempt to take some of it.
- Period of uncertainty - the months it would take to reach a new home would be time in which the colonies could become restless and revolt. A lot of the weight behind power would come from Britain itself. I suspect other European powers would like to spread word that the empire lost its home country.
A position we most want to keep: It is likely that the Empire will see revolts as news of this loss spreads. We want, therefore, for the weight of the monarchy and the power that brings, to sit somewhere the Empire needs to keep.
Want to be central/close to Europe still: The Americas were seen as a new world, separate from the rest of the world and would remove the Empire entirely from the European power play.
A country large enough to fit everyone in: Malta or Gibralta are probably too small.
I can't seem to find any times given for a route between Britain and each of the colonies so going purely off distance I would say Egypt or India (though both routes offer tactical ambush points for pirates). These both satisfy the condition of being positions we want to keep. India was a hub of trade, used to British rule and tactically important for relations in the region. Egypt is home to the Suez canal - a real turning point the world trade.
I would, therefore, choose either Cairo or Bombay as the new capital. Whichever you choose has its own benefits. India has been used to British influence since the East Indies company in 1600 and was much less opposed to further rule than Egypt was. Integration here would have been much easier...however the flip side of this is that, if we choose Egypt India is more likely to stay within the Empire and the new influx of British people could turn the tide of Egyptian rebellion and thus retain the larger empire.
I would choose Cairo, but there are so many factors it really depends how you want to play it.