There is a powerful contrast between the kingdom of Israel ("for Mount Zion lies desolate") and the kingdom of the Lord: "your throne endures to all generations". Much of the Old Testament is contrast between God's eternal faithfulness and the changeable loyalty of Israel, but in Lamentations we instead find a contrast in power, with Israel's faith strongly asserted while the author appeals to the Lord's covenant with Israel. The author is being very honest with the Lord: we are weak, you are strong, we're supposed to be your chosen nation! Where are you?
Christians often feel the same way. We feel abandoned and alone in a sinful world, in our own sinful nature, wondering when the Lord will get around to saving us.
He already has.
We know that the Lord has not "utterly rejected us". He may be "exceedingly angry with us" but his perfect wrath is tempered with perfect love and grace. We have what Israel did not have: the gift of salvation through Jesus Christ. The Lord's everlasting faithfulness has been demonstrated for us in that he sent his own son to us, who died for humanity's salvation even as he was put to death because of humanity's evil.
The author of Lamentations knew that Israel would be saved, but not when. Our salvation has already arrived, and if we have accepted Christ's sacrifice for us, then we are already saved. We don't fell saved; we don't feel enthroned in the kingdom of God; mostly we're just wandering through life doing the best we can, trying to obey God. But that struggle is a response to salvation rather than an attempt to stay alive. None of us know when we will fall asleep, or when Jesus will return, so we share an ambiguous timeline with the author of Lamentations - but it is for the end of struggle rather than the fulfilment of salvation. We are not awaiting the kingdom of God because we are already there. We are waiting for when obedience will no longer be a struggle.