The possibility of random walk movements in the log-returns of exchange rate series as well as the characterization of returns by periods of tranquility and volatility is of interest in financial time series modelling. These features make the analysis of exchange rates to be important in the design of policy, portfolio and risk management whenever international trade and finance is discussed. This paper investigated the level of exchange rates market efficiency and volatility at three market phases of US Dollar-South African Rands and US Dollars-British Pounds exchange rates for the daily periods from 2 January 1995 to 8 June 2016. This paper applied the modified Hurst exponent' and volatility modelling techniques on the series log-returns, absolute and squared log-returns used as proxies to volatility. The results of the modified Hurst exponent techniques on the log-returns and squared log-returns show that the future path of the exchange rates returns in these markets are not governed by their immediate past in the second and third identified phases. Thus, the two markets have been in the state of weak form of efficiency since 2001 to the current transition, with UK exchange rate market more efficient than South African exchange rate market. Using the efficiency and volatility indices to classify the two exchange rate markets as well as the market phases, we observe similar transition in the level of market efficiency in these markets, this which is as a result of trade partnership between the two countries. In terms of market volatility, we cannot make a clearer judgement between the two economies. This paper has repositioned the stability foreign exchange rates of South Africa, a member of Group of 20 (G20) countries, with that of a member of the G7 countries, that is, the UK. To the government, risk and portfolio managers, the findings emanating from this paper indicated that the South Africa's Rand is also stable like the British Pound, and thus, can be traded on the international currency market without fear of excessive loss/gain.