It becomes a root finding problem. Find $u^{n+1}$ such that
$$ u^{n+1} + \Delta t \theta F(u^{n+1},t^{n+1}) = u^n + \Delta t (\theta - 1) F(u^{n},t^{n}).$$
Now, you can multiply by a test function, form your trilinear form $w(\cdot,\cdot,\cdot)$ to obtain the weak problem; Find $u^{n+1}$ in an appropriate space $V$ such that
$$w(u^{n+1},u^{n+1},v) = (u^n + \Delta t (\theta - 1) F(u^{n},t^{n}),v)$$
for all $v\in V$. It is just left to add SUPG term $s(u^{n+1},v)$ to get your final problem; Find $u^{n+1}$ in an appropriate space $V$ such that
$$w(u^{n+1},u^{n+1},v) + s(u^{n+1},v) = (u^n + \Delta t (\theta - 1) F(u^{n},t^{n}),v)$$
for all $v\in V$. You are missing the pressure unknowns, I do not know if it is deliberate and you are working with compressible Navier-Stokes equations or you made a mistake but either way you can find the proper SUPG term to use by looking at the literature.
Since you are solving a non-linear root finding problem, you will have to linearize it somehow. Two popular methods are Newton linearization and Picard linearization.