Jo Fitz-Henry kindly sent me Harriott Lear’s death registration from the Scotland’s People website. She died of “chronic bronchitis” at Cherry Bank, Perth, on 16 July 1859 and was buried in Wellshill Cemetery in Perth.
No members of the family were present as William Newsome, Ekeanor Lear’s husband and so Harriott’s brother-in-law, is mentione as “informant, if out of the House in which he Death occurred.” William, who wrote to Edward to inform him of the death, (and Eleanor?) travelled to Perth for the funeral. Lear was was in England at the time and he also went to the funeral, as he wrote to Chichester Fortescue from 15, Stratford Place on 18 July:
I think I told you that my sister Harriett was ill, & not likely ultimately to recover. The last accounts however, were rather improved: until on Saturday Evening a telegraphic message came to my sister in Surrey, to say she was worse: ― & on the following day a second message told that she had died in the course of the night.
In any case I should not have been able to go to Lady W[aldergrave]’s but as it is I am going off to-morrow morning, to get to her funeral on the following day: ― a long journey, near Aberdeen. [Letters, p. 144.]
Harriott (or Harriett as Edward seems to have preferred) died alone as Lear writes from St. Leonards-on-Sea on 28 July:
My sister’s death was so sudden at the last, that her nearer Scotch friends did not get to see her alive, poor thing. She however wrote a note to another of my sisters, only a few hours before her death, ― merely in these words. ― “Do not be grieved that I am alone: Christ is always with me:” & there is no doubt that she died in complete calm & happiness. What a dreary life hers has been! & yet that of thousands & thousands. “There’s something in the world amiss.”
Jo also sent me a link to photos of the grave of William Newsom and his wife Eleanor, née Lear, in West Norwood Cemetery and Crematorium, West Norwood, London Borough of Lambeth, Greater London:
For more on Edward Lear’s family: “Twenty of Twenty-one”: Edward Lear and His Siblings, part 1 & part 2.
A Blog of Bosh 
Although he told Fortescue he was going to the funeral when writing to him on 18 July, Lear did not in the event go. He wrote in his diary for 19 July ‘Packed. Far too unwell to go to Scotland.
At 2¾ left, & at 4 came down by rail to 119 Marina, St. Leonard’s on Sea. Arranged somewhat ― & dined at the Victoria Hotel.’
Thanks, Stephen, I should have known!