What would happen if a plane lost half of its weight instantly? During the Second World War the RAF had a 10,000 kg bomb, the "Grand Slam", which was dropped from a Lancaster bomber.

I assume that when this was dropped the plane would lurch upwards.

Is there any way of knowing just how big and violent a movement this would have been?

According to Wikipedia, a 'loaded Lancaster' weighed 25,000 kg. I'm not sure if this includes a standard bomb load (approx 6,350 kg) or not, but either way it seems that when the "Grand Slam" was dropped the aircraft would, in an instant, reduce to about half or two-thirds of its previous weight.

Is there any way to estimate how big an upward jump this would cause?
In steady flight, the lift $L$ generated by the wings equals the loaded weight so that there is no vertical acceleration.  Thus, the lift just before the bomb is dropped is
$$
L = m_1g =  245\,250 \;\textrm{N}
$$
where  $m_1=25\,000$ kg is the loaded mass and $g=9.81$ m/s$^2$ is the acceleration due to gravity.

In the time interval just after the $10\,000$ kg bomb is dropped, the lift remains the same until the lift is reduced to again balance the new mass of $m_2=15\,000$ kg. During that transient period, the Lancaster will accelerate upwards at
$$
a = \frac{L-m_2g}{m_2} = \left(\frac{m_1-m_2}{m_2}\right)g
$$
which works out as an upward acceleration of two-thirds of a "gee":
$$
\frac{a}{g} = \frac{m_1-m_2}{m_2} = \frac{2}{3}
$$

As @sanchises pointed out, the steady increase in the upward velocity results in a decrease in the angle of attack, so that the aircraft will again settle in a steady climb.