There is precious little published about them. Them here being the Swedish participants. Rolf-Dieter Müller in his 2007 An der Seite der Wermacht: Hitlers ausländische Helfer beim "Kreuzzeug gegen den Bolschewismus" 1941-1945 of which I have a Swedish translation from 2015 doesn't even mention them. He touches on the fate of Danish and Norwegian participants only lightly, noting that soldiers that had fought with the Nazis were punished alongside other collaborators, and their existence largely expunged from memory. So for the Danish and Norwegian cases, both countries that suffered under Nazi occupation, being former SS members was incredibly stigmatizing, as was being a woman connected with the occupiers or a child of such a union. The important thing to to with us here that Denmark and Norway were directly affected and had public and known processes to "sort it out". Which then to some degree are buried in history. Because the national story is one of heroic resistance not one of how helpless the nation was and the people who decided to take advantage for political, economic or other reasons. Obviously someone who knows Denmark and Norway better would do well to chime in, I'm including this to sort of contextualize the differences between the aftermath.

Also from 2015 incidentally is the book "Hitlers vikingar - Svenska SS-officerare" by Lars T. Larsson. There should also be something published by the author on a British publisher on the subject of Swedish SS volunteers, but I've not been able to locate it. The Swedish participation in the SS was very low, around 200 people. In the book the author looks at the archival material on 24 Swedish nationals (some were of split national heritage and could be said to be German or Norwegian) who managed to become or about to be SS-officers. The answer to your question is yes, and no. No surprise there. There was no one who was punished for joining the SS or the war. The only official punishment they faced when returning was potentially passport violations, as most would not have been allowed to leave the country and therefore left illegally. And the occasional case of espionage where the activities of the persons was seen to have been in acting against the nation. Much of the activities of this small group was part of trying to create a larger organisation of Swedish SS volunteers and providing propaganda or delivering information or sign up as future informants. The authorities interviewed these people if they returned (during and after the war) to ascertain if they had taken part in or might continue to be spies or "politically unreliable" basically. The files from these interviews and other material collected in investigations forms the basic material of Larsson's book. This bears noting as the author himself says the interrogators were mostly interested in current political leanings and names of other potentially interesting people, not so much in wartime crimes and if such information exist in the files it's because the subjects volunteered such information. A number of the volunteers remained convinced nazis and would work at the fringes of politics. However, while officially there are no repercussions that is not to say there weren't social reasons to be careful. Being an official avowed nazi did easily lead to social stigma. One returner was outed as former SS member and was effectively ostracised by fellow workers, others had at least in the years following the war trouble when dealing with officials, and trying to return to active military duty was difficult. It seems the majority of the volunteers had some right-wing inclinations and sympathy for the national-socialism, but at least the Swedes described in the book are by no means all ardent hardline ideological nazis. Usually other reasons play the bigger role, their national-socialist sympathy more informed their choice of where to go. E.g. Finland received many times more Swedish volunteers than the SS did so if you wanted to fight the bolsjeviks you had other options. Then again, as soon as it starts looking bad for the Nazis many of the Swedish volunteers are willing to jump ship as soon as they can, very few remain to the bitter end.

Since there are so few cases they tend to be individual and it's hard to say anything overarching. Some remained active nazis, some renounced former ways and lead quiet lives, some lead dramatic lives and some disappeared. There was as strong incentive to hide, minimize and ignore your SS connections for all involved however, even for those who in more private circles might have been ardent nazis.

I should add this subject has been largely ignored in histography too. It did not fit the "official story" of Sweden during WW2 which emphasizes the saving of Danish jews, but tries to minimize the pragmatic pro-winner neutrality with Nazi-German troop transports, playing for the Allies when they were winning or the shameful repatriation of Estonians to certain doom in Stalin's Soviet Union. It's also not uncontroversial from the diplomatic side, Sweden's neighbouring countries all have axes to grind from the "self-serving" Swedish neutrality. Just take the 2017 book Det svenska sveket 1940-45 (i.e. the Swedish betrayal) by Norwegian author Eirik Veum. The cold war period was one where the story of WW2 was about the "Scandinavian brother peoples" and how 2 of them unfortunately were occupied and the 3rd did what little it could. Swedish SS volunteers did not fit into that story, as little as did the much larger amounts of Danish and Norwegian volunteers.

So yes, in Sweden at lest you could admit to your SS past there was no legal ramifications, but you closed a lot of doors in life if you did. In Denmark and Norway, to the best of my knowledge it was much harder to do so. Excepting of course the neo-nazi and right-wing fringes of society.