The expression to amputate one’s mahogany is a jocular elaboration on to cut one’s stick, which means to take one’s departure. The following definition is from A Dictionary of Modern Slang, Cant, and Vulgar Words (2nd edition – 1860), by the English publisher and author John Camden Hotten (1832-1873):

Cut one’s stick, to be off quickly, i.e., be in readiness for a journey, further elaborated into amputate your mahogany.

—Cf. also the expression mahogany reef.

The expression to amputate one’s mahogany occurs in Copy of a letter from His Holiness Pope Pius IX, a humorous forgery published in Trewman’s Exeter Flying Post (Exeter, Devon, England) of 9th May 1850:

Jamque necesse est mihi facere istos Gallicos (quorum dux “Oudinot” tardus potius est currus per viam) Swieteniam Mahogani¹ suam amputare quod vulgo exprimitur “baculum² suum secare” […].
                                                       (Done into vulgar English)
And now I must make those French fellows (whose General, Oudinot, is a rather slow coach by the way) amputate their mahogany (vulgarly cut their stick).

¹ The scientific Latin Swietenia mahagoni designates the West Indian mahogany; the genus Swietenia was named after the Dutch-Austrian physician Gerard van Swieten (1700-1772).

² The Latin noun baculum, or baculus, meaning a supporting staff, is probably the origin of the English adjective imbecile. This adjective, which originally meant physically weak or impotent, is likely to be from an assumed Latin imbacillus, literally tottering for lack of support.

A variant, to amputate one’s timber, was used for example in an article about County Kilkenny election published in The Independent (Wexford, County Wexford, Ireland) of 21^st August 1847:

There was a bit of a “shindy” in Patrick-street subsequently, between Mr. Butler and Mr. Hely’s body’s guards, after they had accompanied these candidates home. The friends of Mountgarret certainly had the best of it; for the Galmonian Standard-bearers lowered their colors, and, “surrounded by their brethren,” ran “like men,” their opponents for some time giving them hot pursuit. However, on reaching the Parade, a few of the fugitives wheeled round, and discharged a volley of stones at the vanguard of the Mountgarrets. The appearance of the main column of the latter however made the valiant slingers “amputate their timber” again.