The high prevalence of people named Smith in Britain and its former colonies presumably attests to a large population of smiths at some point in time. Likewise, the large number of Müllers in Germany and Switzerland (also Melnik - miller - in Ukraine) would suggest that these places once had more millers than any other profession. If so, why should Britain have more Smiths than Millers, and Germany more Müllers than Schmidts?
My statistics were sourced here: https://mymodernmet.com/most-common-surnames-map/
The cultural factor explaining this is likely just one of regional dialects and spelling preferences. According to the website forebears.io, which takes its information from census data, the surname "Schmidt" is the second most common surname in Germany after "Müller". However, the surnames "Schmitz" and "Schmitt" are the 28th and 30th most common surnames in Germany. If all three are added together, they would outnumber Müllers by a healthy margin (and would still top them if adding in Muellers and Mullers).
English more substantially standardized the spelling of the word Smith. There are spelling variations such as Smyth and Smythe, but neither ranks higher than #660 among most popular surnames in England. The split in spelling variations among German "Smith"s appears to account for the difference.
These kinds of spelling splits likely explain many other discrepancies between the two countries rather than anything about occupational differences themselves. Müller is more common in Germany than Miller is in England - though, interestingly, the discrepancy isn't quite so large between Germany and the United States, likely because of Müllers and other northern European variants being anglicized as Miller upon immigration to the U.S. In England, this can mostly be explained by the popularity of the English surnames "Mills", "Miles", "Mullin", and "Mullins" which essentially mean the same thing as Miller, but which aren't really used in Germany (nor Mülls or Müllen or Mullen or anything else with any notable popularity).
Similarly, Johnson is much more popular in England than Jansen is in Germany, but this can be explained by the fact that in Germany, the surname is also often spelled Janssen, Janzen, Hansen, or Hanssen, due to regional dialects and spelling preferences when surnames became standardized. Hansel and Hanselmann also have the same meaning, whereas the English Johnson doesn't have anything approaching the same split. While there are "Jansen"s in England, many of those will have been German immigrants, and the English surname Johnston/Johnstone is actually a different surname, which is habitational rather than patronymic. It derives from the name of a couple of different places in England historically known as "John's town". There was no German place known as "John's town" that notably gave rise to an equivalent habitational surname (though there are a handful of Johnnasburgs, Jandorfs , and others, and some Hanselmanns may also be derived the same way, but none approaching the popularity of the English Johnston/Johnstone).
Likewise, English "Jones" outnumber German "Jans" but when comparing Jones+Johns vs. Jans+Janz+Hans+Johannes, the discrepancy isn't as stark.
In other words, these common surnames are very common in both countries. The ones that top out the lists tend to be ones with fewer spelling variations in the respective country. The English surname spelling "Smith" is more standard than the German "Schmidt" which has notable competition from Schmitt and Schmitz, while Müller is more standard in Germany as compared to the English Miller which faces notable competition with Mills, Miles, Mullin, and Mullins.