Yes there is a clear political leaning- the audience seems to like anti-Tory jokes and is anti-Brexit but I think most satire over the years has been anti the ruling party. And they do sometimes have a joke about Dianne Abbot to give some semblance of balalnce
It's also repeated at 11.00pm on Monday evenings on Radio 4 Extra.
Don't listen as often as I used to; preferred it when there were more questions per programme which kept the comedy punchy rather than somewhat drawn out on a narrower range of topics.
I have been listening to the News Quiz on and off, but mainly off for many years and it has always been somewhat anti establishment, and long may that continue.
The show really comes alive for me, when Jeremy Hardy is on the Panel.