One stamp and no second stage
The apostille exists so that treaty members can accept each other's paperwork without a second inspection. Britain and Sweden are both inside that arrangement, which is why a stamped British certificate is ready to use on arrival. Nothing is added to the file afterwards and no consular fee falls due.
Sweden's place in the Convention
Sweden joined the Hague Convention in 1999. Since then a British apostille has been the correct and sufficient certification for a document used there, with no consular stamp on top of it. That is worth stating clearly, because a lot of general advice about legalisation describes a second stage this route simply does not have.
Why the EU rule does not shorten this
Many people expect European rules to have removed the stamp entirely. Regulation 2016/1191 does spare certain public documents any legalisation when they cross between EU member states, but a British document is outside that scheme. The exemption does not apply, so the apostille remains the route from Sheffield to any Swedish office.
The British side stays simple regardless. One office covers the country by post and keeps no counter in Sheffield, so the local work is ordering source papers: the Register Office at the Town Hall for civil entries, a university for a degree, a professional body for a registration.
Swedish, and registering after the move
Swedish offices work in Swedish, so a stamped English certificate is normally handed over with a Swedish version beside it. Prepare the words after the apostille is on. A translation made too early describes the plain sheet, while the clerk is looking at a document that now carries a government stamp.
Registration is what most of these files are for. Moving to Sweden means proving identity, marriage and family links to the population register before daily life becomes possible, so birth and marriage entries lead the traffic. Degrees and professional registrations follow for work, and company records appear where a Sheffield firm opens a Swedish office.