Yes, I think the horror stories may be getting to you about the NHS.

Particularly for urgent care, it is quite easy to be seen. I expect your private health insurance won't come into that at all.
As mentioned above, depending upon where you live and what your local surgery's practice does, you will have a variety of options for care. I live in a very rural area now, but when I lived in Exeter, the GP had normal hours with urgent cases seen in reserved slots throughout the day. They also had a doctor 'on-call' all day who would phone you at home. After hours, they had a special out of hours centre. It was based at the hospital but not part of the ER or anything. I used it once after my second son was born and we were in and out within an hour. There is also NHS Direct which is a phone number you can ring for advice.
Some Americans seem to have a bit of trouble getting registered with a GP here but it
shouldn't be a problem.
I'm not sure who your private insurer will be, but they all work roughly the same here. They are meant for treatment of accute but not chronic conditions. That said, your employer may have a bespoke (custom) plan where they can pretty much agree whatever they want. For example, most private insurance here does NOT cover pre-existing conditions (i.e. your UTIs). BUT your employer *may* have an agreement with them that allows for them. So make sure you find out what exactly their plan covers.
That said, I suspect you would not have a problem getting good treatment for your UTIs here on the NHS. Though I'd suspect they'd want to know why you keep getting them. So if you've been thoroughly investigated in the US, you might want to bring your medical notes with you.
HTH!