We consider the statistical inverse problem of recovering an unknown function f from a linear measurement corrupted by additive Gaussian white noise. We employ a nonparametric Bayesian approach with standard Gaussian priors, for which the posterior-based reconstruction of f corresponds to a Tikhonov regulariser f¯ with a reproducing kernel Hilbert space norm penalty. We prove a semiparametric Bernstein-von Mises theorem for a large collection of linear functionals of f, implying that semiparametric posterior estimation and uncertainty quantification are valid and optimal from a frequentist point of view. The result is applied to study three concrete examples that cover both the mildly and severely ill-posed cases: specifically, an elliptic inverse problem, an elliptic boundary value problem and the heat equation. For the elliptic boundary value problem, we also obtain a nonparametric version of the theorem that entails the convergence of the posterior distribution to a prior-independent infinite-dimensional Gaussian probability measure with minimal covariance. As a consequence, it follows that the Tikhonov regulariser f¯ is an efficient estimator of f, and we derive frequentist guarantees for certain credible balls centred at f¯.