Yes, but there may be some leeway (and under the Montréal Convention I don't think there's an exception for returning home either). If - for example - you had your one and only PC power unit in your delayed baggage then I would expect that to be covered as a replacement item, and there was a case here of someone able to claim for an electric shaver (quite why a disposible razor wasn't used temporarily is still a mystery to me, but nevertheless BA paid, and it was a domestic too, albeit with a lot of twists and turns). I also know of a case where BA were unable to return the bags before the next trip - despite it being a week later. BA paid for replacement items there, as one would expect.

I suspect this hinges on why you regard it as "essential" and why you didn't have anything at home suitable. Normally a homecoming isn't regarded as entitling essentials. It wasn't clear (to me at least) in your first message that this was the case either, hence why you were steered in a direction which may have been wrong.