Most government IT projects are given to contractors after a bidding process, the GDS and NHS Digital are teams of skilled developers working within the government.
Amongst others, there's fine work being done at DVLA, DVSA, MOJ, DWP and Home Office Digital are in good shape too
Worked at three of those. Yes contractors, but also staff. Part of one my roles as delivery manager at one of the 'exemplar' projects was to train up my replacement (which was cool). I'm a massive fan.