Sweden takes one certificate
The short answer shapes everything else. To legalize documents for Sweden you need a single certificate from a British desk, and after that the paper is treated as genuine at the Swedish end. No mission looks at it, no mission fee is charged, and no appointment stands between you and a finished file.
So embassy attestation for Sweden is not a stage to plan around, and a quote that lists one is charging for a step nobody performs. What is left is small and easy to control: the right record, a certified copy, the British certificate, and the post between them.
Where a Swedish move actually stalls
Almost every hold-up on this corridor comes from the record itself, not the stamp. A short-form birth entry that omits parents' details, a marriage entry issued years ago in a different format, or a name that appears one way on a degree and another on a passport will each stop a Swedish registration file dead. Consular legalization for Sweden would fix none of them.
Getting the full, current version of each record from the start is the whole trick. Leicester Registration Service can print a fresh certified copy of a city birth, marriage or death entry, and a copy issued now looks the way a modern Swedish file expects, which an old certificate in a drawer often does not.
Before the envelope leaves Leicester
People looking for embassy legalization for Sweden from Leicester are usually working to a start date at a job or a term at a university. Batch the papers: order every copy together, send them to Milton Keynes as one bundle, and you pay one round of postage and wait one turnaround rather than repeating both for each sheet.
Document legalization for Sweden then depends on the post more than on any office. The FCDO desk has no public counter, so a tracked courier is the only real safeguard, and sending a fresh copy rather than the certificate you own means a lost envelope is an annoyance instead of a disaster.