“Our quest is practically finished. I shall call with the King to-morrow, and with you, if you care to come with us. We will be shown into the sitting-room to wait for the lady, but it is probable that when she comes she may find neither us nor the photograph. It might be a satisfaction to his Majesty to regain it with his own hands.”
“And when will you call?”
“At eight in the morning. She will not be up, so that we shall have a clear field. Besides, we must be prompt, for this marriage may mean a complete change in her life and habits. I must wire to the King without delay.”
"Wire" in this case referred to telegraph wire, and to "wire" someone was to send them a telegram. Working models of the electrical telegraph were patented in the 1830s, and commercialization saw the spread of telegraph infrastructure and the improvement of technology, including the laying of undersea cables beginning in the 1850s. The telephone was patented by Alexander Graham Bell in 1876, and wireless telegraphy (radio) was a hot topic in the 1890s, with breakthroughs by Marconi in the 1890s; these latter two appear in the Holmes stories as Doyle continued to write them and technology progressed, but the old "wire" system of telegrams remained an important piece of technological infrastructure through the First World War and into the interwar period.
For more information, I'd recommend Science and Technology in the Age of Hawthorne, Melville, Twain, and James and The Victorian Internet: The Remarkable Story of the Telegraph and The Nineteenth Century's On-Line Pioneers.