In addition to what is already mentioned in the other answers, I would like to point out that you are both overestimating and underestimating the time the transition takes in the US.
The US Presidential Election is an indirect election. In the election on November, 4th, the citizens of the USA did not elect the President. They elected the Electoral College. It is the Electoral College that elects the President, and this vote does not happen until December, 14th. So, if you are simply counting from election to inauguration, then the transition period is actually only five weeks. And the votes are only counted on January, 6th, so if you are counting from the moment that we know who the winner is, then that is only two weeks.
On the other hand, about 1000 of the political appointees require Senate confirmation, and these confirmation hearings can drag on for months! So, if you count from election to having every single member of Cabinet, every agency Director, etc. installed and confirmed, that can require the better part of a year.
The appointees are really the crux here. In the UK and many other countries, the overwhelming part of the government bureaucracy are professional bureaucrats with no political affiliation. Their job is simply to enact the will of the government, whoever that may be. Only the very top level of bureaucracy are political appointees, and only this top level needs to be exchanged.
In the US, however, political appointees go much, much lower into the actual nuts and bolts of the bureaucracy. There are thousands and thousands of posts that will need to be filled, and as mentioned above, up to 1000 of those posts require Senate confirmation. Consider that the Senate will likely have a Republican majority when Joe Biden is inaugurated, and you can imagine that that is not easy.
Not only is it the case that simply filling all those thousands of positions takes very long, there is an additional problem: because this change is so deep, a lot of institutional knowledge is lost every time the administration changes. In order to prevent that from happening, there need to be lengthy handovers between the old and new bureaucrats. In a country like the UK, the same person works in the same department maybe for decades, and then has multiple years to train their replacement. This works to preserve institutional knowledge.
And let's not forget that every one of those multiple thousand appointees may have an entire staff of their own, or at least a secretary or an assistant.
All in all, the really simple answer is: there is just a lot more stuff to do.