If I am looking to get a web application up and running quickly, how important is it to hire a visual designer in addition to a UX designer, or will a good programmer be able to bridge the gap between a good flow and a workable prototype?

You want to hire specialists. Ideally, a developer, UX designer, and visual designer. First and foremost, let's clear up what it means to be a UX designer and a visual designer. A UX designer should not be expected to do graphic design. Yes, UX designers should have a general understanding of visual design , but it is almost impossible to do the stakeholder interviews, journey mapping, card sorting, heuristic analysis, competitive analysis, information architecture, wire-framing, brand development, visual competitive research/analysis, mood boards, design, and production work at the same time and exceedingly well. UX design and visual design are two separate and related disciplines that demand full-time attention. However, let's say you're adamant about finding a person who calls him/herself a UX designer and visual designer. It's not like they're going to have the same rate as a specialist. They're essentially doing two jobs at once, and will probably demand more equity/cash. That being said, you won't save by hiring a designer that does everything. In fact, you may lose - especially on quality.

It depends... If you can create a user friendly interface and don't really care how it looks then more power to you. But keep in mind, you could have the fastest, coolest application on the planet, but if it doesn’t look good, and/or if it isn't user friendly then no one will want to use it. As a designer myself, I have found that you can only really be the best at one thing. I might be a good designer, but I can only program so well... And every programmer I've ever worked with has told me that they really can't design.

I wouldn't rely on your programmer to bridge the gap. However, many uxd have some visual design skills. If you're looking to design something soft on graphically details -- you may be able to get away with not having a visual designer. But in the end, I wouldn't advise it. A visual designer can really bring polish to your product.

A good web designer should have a good grasp on designing for a solid user experience (UX). We normally start with a working prototype/wireframe that addresses a lot UX concerns, and then our designers finesse it when it goes to the creative step. Additionally, we will sometimes do user testing with the prototype or creative to see if there are any issues that arise with common users. Typically a "programmer" won't have the skills necessary to do any of this, and our experience it's best to let them worry about making sure technically everything is solid and functional.