It has been years since I read Mrs. Dalloway, and I don't remember if Septimus Warren Smith's rank was ever indicated.
I'm vaguely left with the impression he was an officer. Was his rank ever identified?
The description of Septimus’ military career does not say what rank he reached, only that “he was promoted”:
Septimus was one of the first to volunteer. He went to France to save an England which consisted almost entirely of Shakespeare’s plays and Miss Isabel Pole in a green dress walking in a square. There in the trenches the change which Mr. Brewer desired when he advised football was produced instantly; he developed manliness; he was promoted; he drew the attention, indeed the affection of his officer, Evans by name. It was a case of two dogs playing on a hearth-rug; one worrying a paper screw, snarling, snapping, giving a pinch, now and then, at the old dog’s ear; the other lying somnolent, blinking at the fire, raising a paw, turning and growling good-temperedly. They had to be together, share with each other, fight with each other, quarrel with each other. But when Evans (Rezia who had only seen him once called him “a quiet man,” a sturdy red-haired man, undemonstrative in the company of women), when Evans was killed, just before the Armistice, in Italy, Septimus, far from showing any emotion or recognising that here was the end of a friendship, congratulated himself upon feeling very little and very reasonably. The War had taught him. It was sublime. He had gone through the whole show, friendship, European War, death, had won promotion, was still under thirty and was bound to survive. He was right there. The last shells missed him. He watched them explode with indifference.
As a volunteer in August 1914, he would have begun as a private in Kitchener’s Army. (That the Armistice found him in Italy means that he must have been in the 23rd Division and fought at the Somme, Messines Ridge, Passchendaele, and Vittoria Veneto.) Promotion would have taken him up through the non-comissioned ranks: corporal, sergeant, warrant officer (after 1915), sergeant major. The 23rd Division suffered about 200% casualties in the course of the war (23,574 killed, wounded and missing on an establishment of about 12,000), so there were plenty of opportunities for promotion.
As regular army officers were killed in large numbers, many volunteers were given commissions. If Septimus had reached the rank of captain, social convention would have allowed him to continue to use the rank in civilian life, but in the novel he is always “Mr.”, so we can be confident that if he was commissioned he was not promoted beyond lieutenant.
However there is no hint in the novel of Septimus being commissioned. If he had been commissioned he then would have been transferred to the command of another unit, whereas he seems to have been subordinate to Evans until the latter was killed just before the Armistice. His close working relationship with Evans suggests that Septimus was the senior non-commissioned officer of Evans’ unit, so I think a good guess is that he reached the rank of company, battalion or regimental sergeant major.