So, I asked my employer, and their response: "The information you received regarding registration in the UK by the University is technically correct – however, our legal team has determined that we will not register in the UK at this time for various reasons. Although, this does not mean that you cannot work in the UK; it just means that since we are not registered in the UK we cannot withhold UK taxes. I do not know about being eligible for UK benefits based on this withholding, so I cannot comment on that portion. As a US Citizen, you are subject to worldwide income withholding and reporting – so we will still withhold Federal, OASDI and Medicare taxes on your income. Once you change you work location to the UK, Illinois State taxes will stop being withheld. You can claim exempt from federal taxes if you would like – but that is up to you and your accountant."
Understood that I'll need to report my UK income and pay appropriate taxes/national insurance in the UK (and similarly need to report in the US). It's clear I'll need to work a qualified professional to get that right. If anyone has personal recommendations in the greater London area, please PM me.
I'm still unclear as to how much risk I'd be taking on vs. that of my employer. Given the responses so far, it seems my obligation is to get my reporting correct, and my employer is taking on all of the risk for being out of compliance with UK employment law. (BTW, the self-employment route makes more sense to me now than when I first started this thread, though the thought of moving to self-employment three months before I have a baby isn't very appealing.)