In addition to Ian's excellent answer, one point to consider is that toddlers are not typically using cell phones as a phone, but instead as a gaming device. This has an important impact on radiation safety: the phone is not held close to the head, but instead is at least a foot away, usually more. As such, they're not meaningfully exposed to the major source of risk.
From the American Cancer Society:
Because cell phones usually are held near the head when being used, the main concern has been whether they might cause or contribute to tumors in this area....
The ACS continues to describe what risks cell phones carry in the main risk case (of being held near the head), and agrees with the WHO:
As noted above, the RF waves given off by cell phones don’t have enough energy to damage DNA directly or to heat body tissues. Because of this, it’s not clear how cell phones might be able to cause cancer. Most studies done in the lab have supported the idea that RF waves do not cause DNA damage.
They then explain some of the major studies, including a few that do appear to show links (and explores why they're controversial, in very good detail from a neutral POV).
The major studies - excerpting the main tagline; please read the full page for a discussion of other lesser findings in the studies, some of which do not directly agree (but have significant limitations, thus why they're not included in the main tagline).
The INTERPHONE study
The 13-country INTERPHONE study, the largest case-control study done to date, looked at cell phone use among more than 5,000 people who developed brain tumors (gliomas or meningiomas) and a similar group of people without tumors. Overall, the study found no link between brain tumor risk and the frequency of calls, longer call time, or cell phone use for 10 or more years.
The Danish cohort study
A large, long-term study has been comparing all of the people in Denmark who had a cell phone subscription between 1982 and 1995 (about 400,000 people) to those without a subscription to look for a possible increase in brain tumors. The most recent update of the study followed people through 2007. Cell phone use, even for more than 13 years, was not linked with an increased risk of brain tumors, salivary gland tumors, or cancer overall, nor was there a link with any brain tumor subtypes or with tumors in any location within the brain.
The Million Women Study
A large prospective (forward-looking) study of nearly 800,000 women in the UK examined the risk of developing brain tumors over a 7-year period in relation to self-reported cell phone use at the start of the study. This study found no link between cell phone use and brain tumors overall or several common brain tumor subtypes, but it did find a possible link between long-term cell phone use and acoustic neuromas.