My neighbour delivered at NUH under a private gynae last year and she said her husband was actually allowed into the operating theatre for her C-section, so NUH does permit it, but she mentioned you should confirm with your specific gynae because apparently it can depend on the doctor and the situation on that day.
Someone in my mum WhatsApp group also mentioned that KKH's no-husband-in-OT policy is quite firm for C-sections, so if having your husband beside you during delivery is really important, that's a big consideration when choosing between the two lor.
I haven't heard specific feedback on Dr Vanaja personally, but generally the NUH private O&G team gets quite good word-of-mouth in my circle! Can I ask, is having hubby in the OT your top priority, or are there other factors like consultation style or fees you're also weighing up? Might help to narrow down which one suits your family better!