I've recently worked through this on my husband's maternal line -- the Swedes originally used patronymic names such as "Anders' son" for the son of a man whose first name was Anders. When these young men went into the military (and perhaps into the merchant marine), there was much confusion because of the duplication -- i.e., too many "Anders' sons"! Many names were changed at that point to reflect their rank, their home place or some geographic feature. My husband's family, for example, went from Andersson to Bjurstrom to Brustrom -- between the military change and the spelling change when they came to the U.S. in the 1890's, it was a real mining job to get back to the right family! I don't know anything about laws/customs re changing names when folks went to England, however.
That said, the Swedes have fantastic documentation in their parish records and a number of really good genealogy sites where you can start by searching for the original family name. You might search for "Swedish naming customs" to get a list of options, for starters.
Good luck! I hope your treasure hunt will be as fruitful as ours has!
