Step 1: The clinical picture of a tonsillar pseudomembrane with upper respiratory symptoms in a child is classic for diphtheria. The bacterium responsible is Corynebacterium diphtheriae, and the question asks which medium yields the quickest result.
Step 2: Speed of growth is the deciding factor. Among standard media, Loeffler's serum slope supports extremely fast multiplication of this organism, with recognisable colonies forming within roughly 6 to 8 hours. This early growth advantage lets the lab report findings before competing flora overgrow the plate.
Step 3: Reviewing why the rest fail: Lowenstein-Jensen (L. J.) medium targets tubercle bacilli and needs weeks, MacConkey's agar selects for lactose-fermenting and non-fermenting enteric rods, and citrate medium is a confirmatory biochemical reagent rather than an isolation medium.
Step 4: Therefore the nasal swab is streaked on Loeffler's serum slope for the fastest confirmation.
\[\boxed{\text{Loeffler's serum slope}}\]