I would expect Congress would declare war rather swiftly, and all attempts to obtain the President's release through military action would be tried. Even if Congress doesn't declare war, the Vice President would be sworn in as President following from Article II, Section 1 of the Constitution and they would take military action:
Article II, Section 1, Clause 6
In Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the Same shall devolve on the Vice President, and the Congress may by Law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President, declaring what Officer shall then act as President, and such Officer shall act accordingly, until the Disability be removed, or a President shall be elected.
The official policy of the government is to not negotiate with terrorists, and this would be seen as a terroristic act. The Constitution combined with the Twenty Fifth Amendment provide clear lines of succession, and Americans have rarely been all that attached to whoever happens to be in power anyway. Sending a message to the offending country would probably be more important than the captured President's life regardless of who the President is.
To answer your more specific questions:
How would be that person be nominated? How long will it take to nominate him/her?
The Vice President would be sworn in probably less than half an hour (give or take) after officials have realized what has happened. All that is really needed is a swearing in ceremony after a quick meeting of the Cabinet secretaries and (possibly) a letter is sent from the Vice President to both President pro tem of the Senate and the Speaker of the House given these facts, if the parties wish to strictly adhere to the 25th Amendment.
Section 4. Whenever the Vice President and a majority of either the principal officers of the executive departments or of such other body as Congress may by law provide, transmit to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office, the Vice President shall immediately assume the powers and duties of the office as Acting President