The clock starts ticking on the day that the judge approves the separation agreement and issues the order of divorce.
The Judgment Divorce Nisi is issued thirty days after the order, and becomes an absolute Judgment of Divorce ninety days after that. So, the answer is that you have to wait at least 120 days from the date of the order before you can remarry.
Let me clarify a little –
You have to wait 120 days after the Final Hearing if it's an uncontested (1A) divorce. You wait 90 days after the Final Hearing or issuance of the judgment if it’s a contested (1B) divorce.
I believe you have asked two questions here:
Question 1. How long does it take for a divorce to be finalized in Massachusetts.
The type of divorce case filed affects how long it will take. An uncontested divorce filed under M.G.L. ch. 208 S 1A usually takes about five months from the date of filing to the date a Judgment of Divorce Absolute is issued (i.e. the divorce becomes officially final). I'd estimate five months because the time it takes for the Court to schedule your uncontested hearing varies. If you have a complete Separation Agreement and both parties are willing to sign an Affidavit indicating they believe their marriage is over, then you can file for a 1A uncontested divorce.
Upon filing the court schedules a hearing (usually 20-30 days after filing). At that uncontested hearing, if the Agreement is approved, the Judge will issue Findings of Fact. The Judgment of Divorce Nisi will issue 30 days later and the Judgment of Divorce becomes absolute 90 days after that. So from the moment of filing until the Judgment becomes final takes about 5 months.
A contested divorce, however, can take much longer. A Divorce filed under M.G.L. ch. 208 S 1 (fault divorce) or S 1B (no-fault divorce) starts with a Complaint for Divorce and indicates that one person is asking the Court to grant a divorce. Under the current time standards the Court's goal is to hold a divorce trial within fourteen months after the filing of the Complaint, however in my experience this ends up being closer to two years. It is still possible, though, to settle a Complaint for Divorce, and most cases do settle without going to trial.
You can settle your case at any time after filing, though if it is within the first six months you will have to convert to a 1A uncontested divorce (with the 4-5 month length described above). If you settle a contested divorce the case will have lasted at least 6 months. Upon presenting the settlement to the Judge or presenting a case at trial, the Judge will issue a Judgment of Divorce Nisi on the date of hearing/trial. That Judgment of Divorce becomes absolute 90 days after that. So from the moment of filing until the Judgment becomes final takes at least 9 months and could take much longer.
2. How long after the divorce can you remarry?
You can remarry once the Judgment of Divorce is final as described above. Once the Judgment is absolute you can remarry.