During the current US government shutdown many federal employees are not working, leading to some services not being provided.
The Customs and Border Patrol (CBP) website states:
Lapse in Federal Funding Impact on CBP Website Operations Notice
NOTICE: Due to the lapse in federal funding, this website will not be actively managed. This website was last updated on December 21, 2018 and will not be updated until after funding is enacted. As such, information on this website may not be up to date. Transactions submitted via this website might not be processed and we will not be able to respond to inquiries until after appropriations are enacted.
From the statement "Transactions submitted vis this website might not be processed" I infer that if I submit an ESTA application there is a possibility that it might be processed. But given how far into the shutdown we are, I suspect that even if they are processing them slowly, there will be a backlog, and it'll be a while before they'll process one submitted today.
The Sun tends to agree with this point of view, but goes as far as to state that noone without a current ESTA can travel:
Donald Trump’s government shutdown means Brits will not be able to visit the US if they don’t already have an ESTA
ESTAs are issued by US Customs and Border Protection, which is among the services affected by the partial government shutdown
However, The Sun is not the most trustworthy of news organisations. The Independent (a trustworthy news organisation) stated during the last shutdown in January 2018:
The ESTA system for online travel authorisation, as used by most British holidaymakers and business visitors, will continue to function as normal; it is self-financing.
and this answer from elsewhere on this site states:
Staff such as TSA, CBP [...] are all considered "essential employees" and thus are still working regardless of the partial shutdown
But just because CBP officers are working at airports doesn't mean they're working in office roles looking at ESTAs...
Does anyone have anything that can clear up this issue? Most likely recent personal experience is the only thing that'll give full clarity. I was tempted to trust The Independent (the article was written by Simon Calder, who's generally acknowledged as the UK travel expert), but then I noticed it's from a year ago and not the current shutdown.
I'm wary of committing money to flights and hotels if it turns out I can't get the ESTA. (And no, I can't just apply to ESTA myself and find out in 72 hours. I need to get a new passport first, which is going to take 3 weeks. And I want to travel in 5 weeks.)