Yes they can get the Christian marriage registered under the Special Marriage Act, 1954.
Section 15 & 16 of the Act provides the complete procedure for this, please read the same:
Registration of marriages celebrated in other forms.—Any marriage celebrated, whether before or after the commencement of this Act, other than a marriage solemnized under the Special Marriage Act, 1872 (III of 1872) or under this Act, may be registered under this Chapter by a Marriage Officer in the territories to which this Act extends if the following conditions are fulfilled, namely:—
a ceremony of marriage has been performed between the parties and they have been living together as husband and wife ever since;
neither party has at the time of registration more than one spouse living;
neither party is an idiot or a lunatic at the time of registration;
the parties have completed the age of twenty-one years at the time of registration;
the parties are not within the degrees of prohibited relationship: Provided that in the case of a marriage celebrated before the commencement of this Act, this condition shall be subject to any law, custom or usage having the force of law governing each of them which permits of a marriage between the two; and
the parties have been residing within the district of the Marriage Officer for a period of not less than thirty days immediately preceding the date on which the application is made to him for registration of the marriage.
Procedure for registration.—Upon receipt of an application signed by both the parties to the marriage for the registration of their marriage under this Chapter the Marriage Officer shall give public notice thereof in such manner as may be prescribed and after allowing a period of thirty days for objections and after hearing any objection received within that period, shall, if satisfied that all the conditions mentioned in section 15 are fulfilled, enter a certificate of the marriage in the Marriage Certificate Book in the form specified in the Fifth Schedule, and such certificate shall be signed by the parties to the marriage and by three witnesses.