The Council on Legislation is the body that checks draft bills before they can be submitted to parliament. Its decisions are advisory and non-binding, but the government often clarifies legislation based on the council's suggestions.
Depending on the council's response, the next step is to put the proposal to parliament, and it is set to come into force on December 1st if it is passed.
After the UK left the EU on February 1st, 2020, it entered a so-called transition period during which UK nationals retained the same rights as before to move to, live in, and work in Sweden. This transition period will end on December 31st, 2020, after which EU law will no longer apply to Brits.
Some general provisions are covered in the Withdrawal Agreement between the UK and the EU, and the proposal, first published in late February, would supplement this agreement. Since February, the proposal has been out for consultation to receive feedback from various relevant agencies.
Under this proposal, Brits would be required to apply to the Swedish Migration Agency (Migrationsverket) for a new residence status (uppehållsstatus), as opposed to simply registering their changed status.
That applies to all UK citizens and their family members who settle in Sweden before the end of the transition period.
To be granted the status, they should fit into once of the following categories: be either employed or self-employed; have sufficient assets and sickness insurance to take care of themselves; be a family member of another person who meets the requirements; or have previously received right of residence (i.e. have legally lived in Sweden for more than five years, which means they no longer need to meet any of the other requirements).
Those who are granted this status will be able to continue living and working in Sweden in much the same way as before.
The government also proposes that the Migration Agency issue documents to cross-border workers (for example, those living in the Skåne region who commute via the Öresund Bridge to Denmark).
UK nationals who move to Sweden after the end of the transition period are currently expected to have to follow the process for applying for a residence permit the same way as other non-EU nationals.
The proposal states that “a fee should not be charged for the issuance of proof of residence status”. The EU left it up to individual member states to decide if a fee would be charged for the permit (although this may not be higher than fees for equivalent documentation such as an ID card or passport), and what the format of the documentation would be.
You can read more about the proposals (in English) here.