What does 'currency convertibility' mean?
This definition of a freely convertible currency may be clearer to you:


  Freely convertible currencies (also called as fully convertible or permitted currencies) are those that anyone can convert into another foreign currency without restrictions or interventions by government of the original country.


It makes clear that the "ease" you refer to typically refers to the lack of government restrictions.

The stereotypical case of a non-convertible currency is one for which the government maintains a fixed exchange rate.  Imagine the UK pegs £1 to be worth \$2, although the market value (i.e. the equilibrium if restrictions were lifted and the goverment didn't intervene) is only \$1.80, say.  In order to maintain this exchange rate, the UK government has to be willing to buy sterling for a price of \$2 from anybody that wants to sell it.

Often this leads fairly quickly to the UK government running short of dollars, at which point the temptation is to try to bring in some sort of restrictions on who can buy dollars, in what circumstances, how long they have to wait for them to be delivered, etc. -- in order to slow the outflow of foreign exchange.  They may also introduce laws that say that when British firms sell goods abroad, they have to convert the dollars they earn to sterling through the central bank, to provide a bigger source of foreign exchange.  They may also introduce laws banning the conversion of pounds by banks, street traders, etc.

By this point the "ease" with which sterling can be converted is pretty low; it has low convertability.

In the example you mention, the UK suddenly removes restrictions, returns the pound to full convertability, and buys dollars from anybody that wants to sell.  In this situation, you'd expect all sorts of people to rush to buy dollars from the UK government at the favourable rate before they run out and are forced to reintroduce restrictions.