What happens if an annulment isn't granted and a couple is remarried and want to become Catholic?
I infer you’re speaking of a scenario in which a divorced and civilly remarried couple has sought annulments from their previous marriages, one or both spouses didn’t receive an annulment, and yet they still seek to become Catholic.
First, did they appeal their decision(s) regarding the annulment(s)? That would be the first step, assuming they are still within the time frame to make an appeal. If not, and they were not counseled that they had a right to appeal a diocesan tribunal’s decision, which would seem rather unusual, they could possibly seek to appeal on those grounds.
In any event, if they do not receive a declaration of nullity on appeal, they could still become Catholic, but according to the teaching of Jesus Christ that the Church upholds, they would have to live as brother and sister, i.e., without conjugal relations as husband and wife (Pope St. John Paul II, Familiaris Consortio 84).
Such a living arrangement would certainly involve self-sacrifice, but with God’s grace, it would be not simply possible but certainly preferable to flouting the Church’s teachings because of the graces they would receive from the sacraments, particularly the Eucharist, confession, and matrimony itself.
However, first things first. Contact a local parish pastor and also the diocesan tribunal to see where exactly they stand regarding the annulment process.