[lathspell]

Lothlorien would fall if Sauron gained the Ring. The World would be in a second Darkness. Glorfindel says this in FotR during the Council of Elrond, when they are discussing whether or not to bring the Ring to Bombadil as a guardian:

'Could that power be defied by Bombadil alone? I think not. I think that in the end, if all else is conquered, Bombadil will fall, Last as he was First; and then Night will come.'

Glorfindel says here that he thinks that Bombadil wouldn't last if Sauron put all his power to him without having the Ring, for Bombadil would still be guarding it.
The fact that he believes he would still be last means that Lorien and Rivendell would have fallen already. So, yes, Lorien and all other elf-places would've fallen, and either with or without help of the Ring.

There is another quote about this topic in the UT, when Gandalf tells his version of 'the Hobbit' to Frodo and the other Hobbits in Minas Tirith. He is talking about the time when Sauron is groing strong again as the Necromances in Dol Guldur, and about his plans at that stage. At that point he says something like:

'I now believe that his original plan was to gain an army big enough and get rid of his greatest enemies first, to gain a force of orcs big enough to destroy Lothlorien and Rivendell.'

Here also someone is speaking as though Sauron wouldn't even need the Ring to overthrow the elf-realms. It was only the attack of the White Council at Dol Guldur that made it more difficult for him to place a direct attack at the Elves' realms.

greetings,
lathspell