We present results for the isovector electromagnetic form factors of the nucleon computed on the coordinated lattice simulations ensembles with Nf=2+1 flavors of O(a)-improved Wilson fermions and an O(a)-improved vector current. The analysis includes ensembles with four lattice spacings and pion masses ranging from 130 up to 350 MeV and mainly targets the low-Q2 region. In order to remove any bias from unsuppressed excited-state contributions, we investigate several source-sink separations between 1.0 and 1.5 fm and apply the summation method as well as explicit two-state fits. The chiral interpolation is performed by applying covariant chiral perturbation theory including vector mesons directly to our form factor data, thus avoiding an auxiliary parametrization of the Q2 dependence. At the physical point, we obtain μ=4.71(11)stat(13)sys for the nucleon isovector magnetic moment, in good agreement with the experimental value and ⟨r2M⟩=0.661(30)stat(11)sys fm2 for the corresponding square radius, again in good agreement with the value inferred from the ep-scattering determination [Bernauer et al., Phys. Rev. Lett. 105, 242001 (2010)] of the proton radius. Our estimate for the isovector electric charge radius, ⟨r2E⟩=0.800(25)stat(22)sys fm2, however, is in slight tension with the larger value inferred from the aforementioned ep-scattering data, while being in agreement with the value derived from the 2018 CODATA average for the proton charge radius.