As it is stated, this property does not hold: indeed consider the sequence $(u_n)_{n \in \mathbb{N}}$ defined for $t \in [0, T]$ by $$ u_n (t) = \sin (2n\pi t) v, $$ where $v \in V$ is a fixed vector. This sequence satisfies the assumptions, but not the conclusion.
Classical results on this topic involve an assumption of the type $(u_n')_{n \in\mathbb{N}}$ is bounded in $L^2 ([0, T], V')$ and imply the strong convergence (see for example Evans, Partial differential equations, 1998, section 5.9.2).