Why are childhood cancers less frequent than infant or adult?
There are 2 main causes of cancer: "born defective" (heritable or genetic problems producing a large predisposition to cancer) and "accumulated damage" (random mutations accumulated over life).

Due to how the UK health system rations treatment (and the recent stories about Charlie Gard being prevented from traveling to the U.S. or Vatican for treatment), it's likely that the government there often "deprioritizes" (complete guess, no idea if they actually do that) infants with severe heritable/genetic problems causing near-immediate cancer. Obviously this means that those cases never live to the 05-09 age bracket.

Cancer deaths peak at 65 rather than 90 or 100 on similar statistical grounds, although late-in-life cancers are almost always due to accumulated mutations  rather than being born defective. Cancer deaths drop off after age 79 because of the relatively large die-offs in the previous few age brackets, in addition to deaths from old age and related conditions.

If there was infinite funding for health treatment and enough hospitals to match, the graph might instead match your expectations as the public would do everything to maximize survival. This means that some of the 00-04 bracket would survive to later age brackets, with the largest cause of death in those cancers being failed treatment.