Not sure if all commonwealth countries use coins with the Queen’s profile on one side, but speaking for Britain and Canada:
What happens when the Queen dies?
Will new coins be minted with the Profile of Charles or his successor?
I am assuming the simple answer is: Yes. But want to be clear.
800 + years of tradition would say YES.
New Zealand coins bear the profile of the current monarch on one side, and yes, (unless the republicans get their way), when Elizabeth II is finally succeded her heir’s portrait will grace our coins.
Given the lfe expectancy of a coin however, it’ll take a while for most of the currency to change – it’s not completely unknown in New Zealand to find amongst your change a coin with King George VI on it (the Queen’s father)… and he died in 1952… before we changed to decimal currency. 
Also, FWIW, the Queen’s profile has been changed over time. When I was a lad the coins all had her profile as of her coronation. The portrait was updated in 1986 to a more mature visage and again in 1999.
Yes. Each country decides separately that they will indeed carry the portrait of the monarch – it’s an element of sovereignty.
But the monarch selects what pose he or she will be displayed in, at wide intervals – Victoria, in 64 years on the throne, changed her portrait twice or three times; Elizabeth in 50 has made about as many. If the mind has prepared dies with the deceased monarch and begun coinage, it usually continues to strike coins with the current dies for the remainder of the year’s issue, then switches. For example, Elizabeth became queen fairly early in 1952, but the mint had begun striking coins with George VI on them, and finished out the year striking George VI coins, then adopted a portrait of Elizabeth for the new 1953 coins. Most 1936 coins bear Edward VIII’s portrait, even though George V died early in the year and George VI took over at the end of the year when Edward VIII abdicated.
There is, however, one exception to this whole thing – Maria Theresia thalers have continued to be struck since 1760 by Austria-Hungary and a couple of its successor nations, for their recognition factor – there are markets and such in portions of Africa where only that coin is acceptable money for large sums.
The tradition is that the direction of the profile alternates with each monarch. Elizabeth is shown in right profile, so Charles will face left, same as his grandfather, George VI, did.
Although, (utter trivia), George VI’s predecessor Edward VIII faced left (rather than right) on the few coins minted with his image as he preferred his left profile.