Is there no salvation among any Christ-professing faiths, apart from the restored gospel of the Latter Day Saints? (LDS perspective)

“Salvation” and “saved” are used in the scriptures in multiple contexts with several different meanings.

In one context, all of God's children will receive Salvation from Physical Death. Whether they are LDS, of some other Christ-professing faith, or any other faith - including atheists. Every person born to earth has kept their first estate, will be resurrected, and will live on the Earth after it has received it's paradisiacal glory^†. We were saved before we were born, our births are a testament of this, we have all been saved from physical death.

In another context, there is Salvation from Sin. To be cleansed from sin through the Savior's Atonement, an individual must exercise faith in Jesus Christ, repent, be baptized, and receive the gift of the Holy Ghost (see Acts 2:37-38). This is only possible through membership in Christ's restored church, which is the Church that bears his name and his priesthood, the Church of Jesus Christ of Latter-day Saints.

Christ revealed to the Prophet Joseph Smith in 1820 that not one Church on earth had the proper priesthood authority to baptize in his name, that the priesthood had been lost during the great apostasy. That authority was restored through Joseph Smith, and the restored Church is the only church through which a man can be baptized and receive Salvation from Sin.

What is meant by "first estate" and "second estate"?

It's the difference between Salvation, and Exaltation.

To be brief, we all lived as spirits in the presence of God as Spirits before the creation of the Earth. We refer to this time as the preexistence. When God presented the plan of salvation there were two who offered to fulfill the role of Saviour on earth. The first was Jesus Christ, the second was Lucifer. They presented their plans for redeeming mankind and The Father chose Christ's plan over Lucifer's. Lucifer was angry, rebelled against God, and led many spirits away when God cast him out of his presence (see Abraham 3:22-28). Those who chose to stay and accept Christ's plan where those who kept their first estate, and would be privileged to be born on earth, receive a physical body, and be resurrected. Those who followed Lucifer (Satan) did not keep their first estate, and shall remain as spirits forever and dwell in eternal darkness after the earth has received it's paradisiacal glory.

Those who have kept their first estate have been born on earth or will be born someday and face the trials of mortality. Those who are baptized and have receive the Holy Ghost through the proper priesthood authority are conditionally saved from sin. In this sense, salvation is conditional, depending on an individual's continuing in faithfulness, or enduring to the end in keeping the commandments of God, also referred to as keeping their second estate. Those who keep their second estate shall receive Exaltation, and be exalted with our Saviour Jesus Christ and our Father in Heaven.

Abraham 3:22-28

22 Now the Lord had shown unto me, Abraham, the intelligences that were organized before the world was; and among all these there were many of the noble and great ones;

23 And God saw these souls that they were good, and he stood in the midst of them, and he said: These I will make my rulers; for he stood among those that were spirits, and he saw that they were good; and he said unto me: Abraham, thou art one of them; thou wast chosen before thou wast born.

24 And there stood one among them that was like unto God, and he said unto those who were with him: We will go down, for there is space there, and we will take of these materials, and we will make an earth whereon these may dwell;

25 And we will prove them herewith, to see if they will do all things whatsoever the Lord their God shall command them;

26 And they who keep their first estate shall be added upon; and they who keep not their first estate shall not have glory in the same kingdom with those who keep their first estate; and they who keep their second estate shall have glory added upon their heads for ever and ever.

27 And the Lord said: Whom shall I send? And one answered like unto the Son of Man: Here am I, send me. And another answered and said: Here am I, send me. And the Lord said: I will send the first.

28 And the second was angry, and kept not his first estate; and, at that day, many followed after him.

† ^{There are few exceptions who will not live on earth, but instead will be cast out to dwell in outer darkness with Satan and those other fallen angels. They are those who deny the Holy Spirit, which is not something the average man can do. Very few have been identified as committing the unpardonable sin. One we know of is Cain, who slew Able, another is possibly Judas Iscariot. Both men knew God, walked in his presence, and denied him to his face.}