To my knowledge, no private hospital has A&E facilities. However, plenty of private hospitals have the ability to cope with complications during or after surgery.
Not suggesting that the particular Nuffield in question would be able to cope, but you can’t draw a conclusion on that from the absence of an A&E.
An NHS patient has a lot less choice over who performs the surgery, that’s one of the perks of opting for private. That same perk extends to patients who use the e-Referral system to go to a private hospital for NHS funded treatment. The vast majority of NHS funded treatments in Private facilities are reimbursed at (or below) the National Tariff which is freely available.
As I think someone pointed out, the NJR will tell you which hospital performs the most operations in quite a broad category. This may or may not be a good way of picking your hospital.