Note that this answer contains a fair amount of conjecture.
I think that the custom to have the currency sign precede the amount may have something to do with the pre-decimal monetary system in the UK. Before 1971, prices were expressed in pounds, shillings, and pence - or the £sd system. (the d stands for denarius, a small Roman coin). There were 20 shillings in a pound, and 12 pence in a shilling.
The Academic Writing in English wiki of the University of Hull notes that: "When actual sums of money were being written, by the twentieth century it was usual to write the three units separated only by slashes (called solidus by old printers for this reason) after the £ sign: £20/10/6 meant 'twenty pounds, ten shillings and six pence';"
I would postulate that in this system having the £ sign precede the values was the most intuitive method of notation. At least to me, the alternative of 20£/10/6 is far less appealing and even confusing to look at. Placing the £ at the end would not make sense as it would imply that the £ referred to the number of pence. Placement of the £ sign before the numbers clearly signalled that the following three numbers referred to monetary values.
Of course the $ sign was already in use by the end of the 18th century and I'm unsure how monetary amounts were written down at that time in the UK. However, it suspect it must have been similar to the above-mentioned notation as it had already become customary to have the £ sign precede the monetary value in 18th century England. In fact, Florian Cajori, in his book A History of Mathematical Notations (page 26), writes that: "It is interesting to observe that Spanish-Americans placed the ps after the numerals, thus 65ps, while the English colonists, being accustomed to write £ before the number of pounds, usually wrote the $ to the left of the numerals, thus $65." (The use of ps here refers to pesos; and the imposition of the "s" over the "p" is believed to have led to the creation of the $ sign.)
Notice board displaying the entry prices for the Theatre Royal, Drury Lane, London, around 1946 (source: Wikipedia).