The only British monarch whose face has ever appeared on a Bank of England banknote apart from Queen Elizabeth II is George V, who was pictured on the 10 shilling notes printed 1918-1928. The design returned after that to one without the monarch's head on it until 1961. The same was true of the £1 note at approximately the same period. (So Rollo is incorrect - George VI's head has never appeared on a British banknote.) Other notes were basically black copperplate printing on white paper until the early 1960s when coloured designs started to be used - people of a "certain age" still remember the "white fiver" and how, when money was worth a lot more than it is now because of inflation, breaking into a £5 note was quite an event! Pictures of all notes going back for the last 3 centuries are at the Bank of England's interactive web page linked below.
No doubt when Queen Elizabeth II dies, a suitable portrait of the new king will be commissioned and used on banknotes printed after that. Old ones will continue in circulation until they wear out and are withdrawn and pulped, and as paper wears out quickly this is a fairly quick process - the average life of a banknote is two years.
Coins are another matter - the monarch's head has appeared on them for many centuries and traditionally, it changes the direction it faces with each new monarch. Elizabeth II faces right on British coins, so when Charles becomes king, he will face left. Coins can last for many years so it is common for coins of the previous monarch to continue in circulation well into the reign of the next one. New coins will of course bear the head of the current monarch. There are currently no coins in circulation in the UK that don't have the head of Elizabeth II on them - this might have been the case anyway as she has been Queen for 57 years, but another factor is that the UK changed its currency in 1971 - on 15 February that year, it changed from £1 = 20/- (20 shillings) = 240d (240 pence) to £1 = 100p. This meant a massive changeover of coins and a lot of confusion for older people, though one thing that eased the burden on the Royal Mint was that some old coins had exact "new money" equivalents - so that the old 6d, 1/- and 2/- coins carried on for a while as they exactly equalled 2 1/2 p, 5p and 10p respectively.
Which brings me on to what Rollo and Richard are talking about - what Richard remembers is 2 shilling coins, which carried on in circulation after the decimalisation of 1971 for some considerable time as they had the same value as the new 10p. In the same way, the 1 shilling coin continued in use as the 5p coin. In fact the 5p and 10p were made the same size and weight as the coins they replaced so they could be easily identified and used in vending machines without machines having to be altered. The 5p and 10p have now been made smaller and lighter and at that point, any remaining 1/- and 2/- coins in circulation were positively withdrawn as they came into the hands of the banks.
Just as an interesting side note, the UK is the only country in the world not to have its name on postage stamps as it was the first country in the world to issue them and the monarch's head identifies them. Unlike coins, the monarch always faces left on stamps.
Now if the UK ever agrees to abolish the pound and enter the euro, the Queen's head will disappear from banknotes as they have a common design across the "eurozone", but each country that uses the euro still produces its own coins. They have to produce them all to the same specification except that they can choose their own design for the "heads" side - and of course all the monarchies use the monarch's head. So that part would continue. I find it fascinating when I visit continental Europe as I so often do - you can get all kinds of designs in your change!