They can provide you with a large supply of all the equipment you need over a few months.
This is critical as it WILL take a long time to get what your equipment once you have moved.
While they do not know much about the procedures on the other side of the pond, this is something that they can do.
You need to let Försäkringskassan know about your move, but only if moving there for a long time and do no longer consider Sweden your "home".
If you know you will only spend a few months at a time in the UK, this step is not necessary.
Read more on this here.
Same goes here - let Skatteverket know that you have officially moved aborad and intend to live outside of Sweden for a year or longer. You will remain a Swedish citizen and keep your "personnummer" and passport.
You will from time to time have to state when you officially moved, for example to renew your passport. This information can be found once you login in to Skatteverket.
But do this only once you have physically moved. You need a UK address to do this, and given potential housing scams it is definitely worth it to wait a little bit once you have settled in.
More information about this can be found here.
How to deal with your personal insurance is up to you and what your specific insurance company offers for someone who has moved abroad.
When I studied in Glasgow I was able to go back to Sweden often enough to justify a small extension to my insurance for essentially 'travels abroad'.
However, now that I have officially emigrated from Sweden I no longer hold an insurance in Sweden.