I'm no expert, but Find A Grave and the Texas Pensions were searched using many variants of initials, names, and dates you provided. These are the only two Pittmans which came up:
(1) Nathan Lawson Pittman
DEATH 30 Dec 1934 (aged 89–90)
BURIAL Holloway-Pittman Cemetery, Clarksville, Red River County, Texas, USA
MEMORIAL ID 30364476
There is a picture of a CSA Veteran Headstone for Nathan Pittman in Co. H, 1 AL Inf.
MARRIAGE: N. L. Pittman married Della S. Neathery on 25 Mar 1884 in Red River Co TX Book E page 411
Bagwell is about 8 miles northwest of Clarksville, Red River County, Tx
(2) Nathan Pittman
BIRTH 16 Dec 1844
DEATH Jun 1920 (aged 75)
BURIAL Hopewell Cemetery, Swan, Smith County, Texas, USA
MEMORIAL ID 66000358
Nothing at the memorial indication he was a veteran
(1) Texas pension #16296 shows N. L. Pittman of Red River County, Texas was in Company B, Light Artillery in Waul's Legion and surrendered at Vicksburg. In 1864 he was then in McMahon's Battery.
(2) Texas pension #24329 shows Nathan Pittman of Smith County, Texas was in Co H of the 1st Alabama Infantry. Neither his Masonic headstone or his pension shows any indication of a middle name or initial.
It appears some one did not do their due diligence when applying for a veteran's headstone for their ancestor. Nothing for Noel Lemuel Pittman of Timmon's Rgt.
Hope this helps more than it creates confusion.