Details

1839:
the congress of the "republic of texas" allocated land to be used for "two colleges or universities".. "nothing more was done by the Congress or by the Texas legislature until 1858"..

1858:
the legislature finally appropriated the land, along with $100,000 in bonds. "Secession and the Civil War prevented the act of 1858 from being carried out, however."

1871:
legislature established texas a&m college.., "but the university was still postponed."

1876:
the constitution specified that the legislature should "as soon as practicable" establish the university of texas:

The Texas Constitution: Article 7 - EDUCATION: Section 10 - ESTABLISHMENT OF UNIVERSITY; AGRICULTURAL AND MECHANICAL DEPARTMENT: The legislature shall as soon as practicable establish, organize and provide for the maintenance, support and direction of a University of the first class, to be located by a vote of the people of this State, and styled, "The University of Texas," for the promotion of literature, and the arts and sciences, including an Agricultural, and Mechanical department.

"though no tax was to be levied and no money appropriated out of the general revenue for...buildings".

1881:
"on sep 6th, austin was chosen for the site of the main university.."

1882:
"nov 17, the cornerstone of the west wing of the first main building was laid.."

1883:
"The university was formally opened in the new building on September 15, 1883, though classes were held in the temporary Capitol as late as January 1884."

1899:
"The university's first building, known as the Main Building...completed in 1899."

1967:
"On March 6, 1967, the Sixtieth Texas Legislature changed the official name of the main university to University of Texas at Austin."

Summary

"In the first year, 1883-84, the University (of Texas) faculty was composed of eight professors, four assistants, and the proctor. Enrollment for the first long session was 221." This, as opposed to the University of Tennessee, which aside from being a college as early as 1794, was a full fledged university in 1840 with a faculty of "five or six", and a student body of "around one hundred". So the moral of the story is that the University of Texas was "named" in the constitution of 1876, but did not open until 1883. The University of Tennessee was operating in 1840, though under a different name, and was officially renamed "University of Tennessee" in 1879. Since Tennessee was operating under the name "University of Tennessee" four years before the University of Texas was in operation, the designation "UT" should belong to Tennessee.

Further, the University of Tennessee is still called just that, and rightly abbreviated "UT", while the Texas university has since been officially renamed to "University of Texas at Austin" in which case should be abbreviated "UTA" (or UTAU if Arlington won't change theirs). Incidentally, try seeing where www.ut.edu takes you.

Lastly, "orange and white chosen as UT(exas) official colors, 1900", while in "1892 (Tennessee) students endorsed the colors at a special meeting called for the purpose, but two years later were dissatisfied with the choice and voted to drop the colors. after a heated one-day debate no other colors proved satisfactory, so the students returned to orange and white." So at the very least Tennessee has undeniable claim to orange and white.

References

http://web.utk.edu/~mklein/brfhist.html
http://www.tsha.utexas.edu/handbook/online/articles/view/UU/kcu9.html
http://www.capitol.state.tx.us/txconst/sections/cn000700-001000.html
http://www.cah.utexas.edu/exhibits/TraditionsExhibit/page1.html