[edited] I think the order of sacraments you will receive will be the conditional Baptism, followed by Sacrament of Reconciliation, with a conditional absolution.
If your baptism in the previous church was valid, then the Sacrament of Reconciliation will be valid, and that will forgive your mortal sins, and your conditional baptism will be merely symbolic and provide no additional graces.
If your first baptism however was defective, then the sacrament of reconciliation will be unnecessary, as your conditional baptism will be a true baptism, and will indeed wash away all your sins, as well as the temporal punishment due to those sins. You would be in a supreme state of grace, and reconciliation would simply contribute to these graces.
Either way, the conditional baptism will include all the special blessings offered to a Catholic infant or catechuman at baptism, and you will benefit from these additional prayers that were missing from the original baptism.