One big difference is that in the UK the cabinet is made up of MPs, and the second biggest party in parliament has a shadow cabinet at all times; so when they get elected (and, although things may be different next election, UK has traditionally been a two party system so it's always either the current biggest or the second biggest party who will win the election) they basically have the heads of department already in place, ready to take over and start working with the large non-partisan civil service.
Compared to the US, where congress/senators (equivalent to our MPs, in that they're the politicians who just won their local elections) haven't spent the previous years working as "shadow secretary of health/defense/education/whatever" (and they're not the people those jobs will be filled by) and although the president elect may have put some thought into it before the election, they haven't yet finalised decisions and discussed with their candidate for those positions about actually hiring them.
And I'm not sure about the numbers but I think that the US has a larger amount of partisan civil servants (west wing staff and thousands of other presidential appointments across the various departments) compared to the UK where there are partisan advisers for government members, but a larger proportion of work/responsibilities rest with career civil servants who don't change when a new government gets elected.
There's also the potential for slow recounts or, in some states, the need for runoffs (aka a whole new election), which is why there's a gap between the public election and the Electoral College voting - if it were up to me, I'd definitely get rid of the whole concept of the Electoral College, it's ridiculous (see https://news.ycombinator.com/item?id=45072350 & my reply to it).
(I've simplified a few things, for example UK cabinet doesn't have to be entirely MPs it can also include peers from the House of Lords, such as when the current Labour government got elected the PM decided not to keep the MP who had been his Shadow Attorney General in that role, giving it to a Lord instead... but this comment is already too long, so I've left out various details / edge cases like that.)
