Suppose that we have a piecewise-linear one-dimensional basis. Technically the gradient has no point value defined at the finite element nodes because the derivative is discontinuous at the nodes as you've noticed.
However, there are several techniques to approximate the gradient. One example is based on projecting the weak derivative onto a piecewise-linear basis. Let me try to explain it with the following example:
Above is a finite element solution to the problem $-u''=1$, $u(0)=u(1)=0$. Because the solution is piecewise-linear, the derivative is piecewise constant:
This derivative has no unique value at the nodes because it's discontinuous. However, we can perform $L^2$-projection of the above function onto the piecewise-linear basis to arrive at:
Let $u_h \in V_h$ denote the piecewise-linear solution. In practice, the above $L^2$-projection means that we solve for $w_h \in V_h$ satisfying
$$\int_0^1 w_h v_h \,\mathrm{d}x = \int_0^1 \frac{\partial u_h}{\partial x} v_h \,\mathrm{d}x$$
for every $v_h \in V_h$. This reduces into a matrix system
$$M \boldsymbol{w} = C \boldsymbol{u}$$
where $M$ is the mass matrix, $C_{ij} = \int_0^1 \frac{\partial \phi_j}{\partial x} \phi_i \,\mathrm{d}x$ and $\boldsymbol{u}$ is the original solution vector.
