O exército de Sauron alcançaria as Terras Imortais se o Anel não fosse destruído?

Não, as Terras Imortais foram removidas das esferas do mundo quando os Númenorianos tentaram alcançá-las no final da Segunda Era. Isso levou os Valar a convocar Eru, que removeu as Terras Imortais, resultando no afundamento de Numenor:

Then Manwë upon the Mountain called upon Ilúvatar, and for that time the Valar laid down their government of Arda. But Ilúvatar showed forth his power, and he changed the fashion of the world; and a great chasm opened in the sea between Númenor and the Deathless Lands, and the waters flowed down into it, and the noise and smoke of the cataracts went up to heaven, and the world was shaken. And all the fleets of the Númenóreans were drawn down into the abyss, and they were drowned and swallowed up for ever. But Ar-Pharazôn the King and the mortal warriors that had set foot upon the land of Aman were buried under falling hills: there it is said that they lie imprisoned in the Caves of the Forgotten, until the Last Battle and the Day of Doom.

But the land of Aman and Eressëa of the Eldar were taken away and removed beyond the reach of Men for ever. And Andor, the Land of Gift, Númenor of the Kings, Elenna of the Star of Eärendil, was utterly destroyed. For it was nigh to the east of the great rift, and its foundations were overturned, and it fell and went down into darkness, and is no more. And there is not now upon Earth any place abiding where the memory of a time without evil is preserved. For Ilúvatar cast back the Great Seas west of Middle-earth, and the Empty Lands east of it, and new lands and new seas were made; and the world was diminished, for Valinor and Eressëa were taken from it into the realm of hidden things. (The Silmarillion)

Após essa mudança no mundo, ela só era acessível pelos Eldar explicitamente:

the Eldar were permitted still to depart and to come to the Ancient West and to Avallónë, if they would. Therefore the loremasters of Men said that a Straight Road must still be, for those that were permitted to find it. And they taught that, while the new world fell away, the old road and the path of the memory of the West still went on, as it were a mighty bridge invisible that passed through the air of breath and of flight (which were bent now as the world was bent), and traversed Ilmen which flesh unaided cannot endure, until it came to Tol Eressëa, the Lonely Isle, and maybe even beyond, to Valinor, where the Valar still dwell and watch the unfolding of the story of the world. (The Silmarillion)

Eles só eram alcançáveis pelos Elfos e aqueles que recebem isenções especiais pelos Valar, isto é, os Portadores do Anel (e Gimli), embora estes ainda eventualmente morram nas Terras Imortais:

But the promise made to the Eldar (the High Elves – not to other varieties, they had long before made their irrevocable choice, preferring Middle-earth to paradise) for their sufferings in the struggle with the prime Dark Lord had still to be fulfilled: that they should always be able to leave Middle-earth, if they wished, and pass over Sea to the True West, by the Straight Road, and so come to Eressëa – but so pass out of time and history, never to return. ...

But in this story it is supposed that there may be certain rare exceptions or accommodations (legitimately supposed? there always seem to be exceptions); and so certain 'mortals', who have played some great part in Elvish affairs, may pass with the Elves to Elvenhome. Thus Frodo (by the express gift of Arwen) and Bilbo, and eventually Sam (as adumbrated by Frodo); and as a unique exception Gimli the Dwarf, as friend of Legolas and 'servant' of Galadriel.

I have said nothing about it in this book, but the mythical idea underlying is that for mortals, since their 'kind' cannot be changed for ever, this is strictly only a temporary reward: a healing and redress of suffering. They cannot abide for ever, and though they cannot return to mortal earth, they can and will 'die' – of free will, and leave the world. (Letter 154)

Nenhum não-elfo que não tenha "desempenhado um papel importante" em questões élficas seria capaz de encontrar a Estrada Reta que ainda leva a Valinor.

Gandalf, o Cinzento, veio de Aman para a Terra Média. Então, se a passagem dependesse apenas do poder, Sauron certamente poderia forçar a estrada reta e viajar para Aman.

Mas Sauron não sobreviveria por muito tempo em Aman. Sauron é um Maia (menor potência) e seria atendido pelo Valar (poderes maiores) que residem lá.

Então Sauron seria bem aconselhado a esperar até o Fim dos Dias. No Dagor Dagorath , Morgoth (o mentor de Sauron) descobrirá como quebrar a Porta da Noite e destruirá o Sol e a lua. Parece improvável que a estrada reta seja um obstáculo para Morgoth.

would Sauron's forces be able to reach the Undying Lands and conquer every corner of the Earth, not just the Middle Earth?

Conquiste? Considere que Morgoth fugiu das terras imortais e nunca mais retornou com um exército, apesar de ele e seu exército serem muito mais strongs do que os de Sauron. Com uma boa causa: na Guerra da Ira, todo o seu exército e o próprio Morgoth foram derrotados, provando que os Valar eram os mais poderosos. Nem Morgoth nem Sauron combinam com Tulkas, ring ou não. Considere que a totalidade de Arda era considerada o Anel de Morgoth e ele foi derrotado.