How long did it take news to travel from Britain to its far way colonies (such as India and Cape Colony) in the latter half of the 1800s? How did news and messages typically travel?

Well by 1870 India was reliably connected to the UK by telegraph, with much of the rest of the British colonies in Asia and Australia following thereafter.

Before that, in the 1850s mail would have travelled by ship.

But even in 1852, when steam and iron had been long applied to create regular lines of ocean navigation the time-scales for communication were immense. A letter from London took 12 days to reach New York in the United States; 13 days to Alexandria in Egypt; 19 days to Constantinople in Ottoman Turkey; 33 days to Bombay in India; 44 days to Calcutta in Bengal; 45 days to Singapore; 57 days to Shanghai in China; and 73 days to Sydney in Australia. From this state of affairs, within twenty years, the telegraph companies created links that allowed messages to be sent to all of these places in minutes.

Non urgent communications would still have gone by ship.

Quoted from Steven Roberts' History of the Telegraph Companies in Britain. The late Mr Roberts appears to have been an enthusiastic amateur engaged in primary research on the history of telegraphy. His exhaustively detailed text and list of sources are available here