The pillow sword nomenclature goes back at least as early as the Wallace catalog of 190-(1?).
While doing some other work and while having that site open I came across
The insignia include two swords, a cap of maintenance, four serjeants' maces, chains and badges for the mayor and sheriff, and an oar for the water bailiff. The charter of 1483 provided for a sword to be carried before the mayor in the same manner as in other boroughs and cities. (fn. 39) A sword had presumably been acquired by 1486, when the office of sword bearer was mentioned, (fn. 40) and the city had two swords by 1560. The principal sword, which was redecorated to mark the visit of Elizabeth I in 1574, (fn. 41) was lost in the 19th century. (fn. 42) The other sword, perhaps the first acquired, was known as the mourning sword in 1584. (fn. 43) It is 3 ft. 11½ in. long and has been painted black, retaining the original blade and hilt with curved quillons. A third sword, made for the corporation in 1567 and given a red scabbard, (fn. 44) had become the principal sword by the mid 17th century and was depicted on the monument to John Jones (d. 1630). (fn. 45) It is 4 ft. 3½ in long and retains its original blade and hilt. With the scabbard it was altered in London in 1652 to carry the Commonwealth arms. In 1660 those were replaced by Charles II's arms and the scabbard was partly redecorated, some royal badges being reinstated soon afterwards. (fn. 46) A cap of maintenance, recorded on the arms granted to the borough in 1538, (fn. 47) was replaced several times. (fn. 48) It was identified, questionably, as the sword bearer's hat by the mid 19th century and until 1933 when W. L. Edwards, the mayor, gave the city a new cap of maintenance. (fn. 49)
'Gloucester: Arms, seals, insignia and plate', A History of the County of Gloucester: Volume 4: The City of Gloucester (1988), pp. 368-371
"Was known as the mourning sword in 1584" seems more than just speculative and a later descriptive.
In other searches for pillow sword, I agree that does seem to arise as a descriptive in the 19th century but in plowing through some other book searches (ie Google and the site linked above) mourning swords in the context of smallsword types is referenced in a 2006 journal I see only a snippet of but
Society for Army Historical Research (London, England) - 2006 - Snippet view
MOURNING SWORD. I have just completed a translation of Charles Wesley's Journal 1736-38, but there are still a few remaining unanswered questions. One concerns the expression 'mourning sword'. Here is the context: 'April 24...
the context listed 1736 in that snippet view of the said journal.
I have only passing knowledge of the prevalence for mourning swords in Victorian England but the trend seems to have continued not to just blacken swords for the purpose but swords made specifically for the purpose. Some looking quite toy like with the hilts black as made that way and the beaded chain guards.