So in this scenario, Virginia is basically like Kentucky in 1861, but more aligned to the Confederacy. Whereas Kentucky was likely to bolt to the Union when forced to decide, Virginia is likely to bolt to the Confederacy.
You will have volunteers leaving the state going to both sides, but more fort he Confederacy. Lincoln is likely to avoid antagonizing Virginia further, but someone from the south is likely to march in troops. At that point, either Virginia sides with the Confederacy or the Union.
I honestly don't see this independent status lasting long. But if we are supposed to treat this as if Virginia never involves itself with the war, then the Confederacy is probably defeated before the end of 1863. Tennessee quickly falls to the Union as does the Mississippi River Valley, and the Union advances deep into Arkansas and Mississippi in 1862. In 1863, Alabama is invaded and Atlanta taken. At which points some kind of peace is reached.
Virginia just provided too much support for the Confederacy for its absence to not have a huge impact. It was a large manpower reserve, provided its best military leader, and was a significant economic engine. While some of these benefits will still be present to the Confederacy, the lack of full support will be noticeable.
I don't see West Virginia seceding from Virginia. I don't know whether the Emancipation Proclamation takes effect, or if anti-slavery amendments are passed. Given the quickened collapse of the confederacy in this scenario, Lincoln certainly wouldn't need to do it to forestall European recognition. However, much of northern public opinion will be against it. Still, with such quick successes, Lincoln might feel emboldened to do it anyway, confident that it won't hurt the war effort. The only problem is whether concern about Virginia rejoining ultimately prevents that from happening.