This is the best public access point I have found:
Take Interstate 80 west from Salt Lake City to the Tooele Exit #99. Go south on highway 36 several miles to Tooele. At the first intersection with a stoplight (1280 Street), turn left. Immediately make another left (Pine Canyon Road) as it heads back north paralleling highway 36. This road soon heads east directly to the Oquirrhs. After a few miles is the small settlement of Pine Canyon. Turn left on Churchwood Drive, follow to the end where there is room for several cars. Between June 1st and October 31st, the gate should be open and you can drive to the base of the mountain. If conditions are good, the gate will remain open until December 5th.
From this access point, Nelson Peak could be climbed from a number of routes. Taking this route up the western ridge allows you to summit a smaller neighboring peak I call "Nelson Ridge" or Peak 8100. Getting to "Nelson Ridge" is an enjoyable climb, I can't say the same for the traverse over to Nelson Peak.
From the end of the pavement and gate on Churchwood Drive, take the dirt road to the right as it heads east along a property line. Eventually it will reach a north/south powerline road. Follow it north along the powerlines until you reach another gate. Beyond is private land but no signs showed it and most wouldn't know this. Just past the gate is a road to the right entering Flood Canyon. Inside Flood Canyon, somewhere near the end of the road, you are back on BLM land. (We ditched our bikes here...a 2 mile ride from the Churchwood Drive Gate. From the area near the end of the Flood Canyon road, climb north up a drainage until gaining one of several ridges accessing the long west ridge of Nelson Peak. Reach point 7870, the far west point on the ridge to Nelson. There is some brush below this point but game trails are excellent and most of it can be avoided. After a few humps, you reach the summit of "Nelson Ridge". Beyond is a 400 foot drop, then a ridge walk that is often brushy with rocky outcrops that must be negotiated around. Maybe in summer this wouldn't be that bad but with fresh snow it was an awful snowshoe for us. The ridge between point 8497 and Nelson was open and enjoyable. The final 100 feet up Nelson is blocked by a cliff band with intermittent sections you can get climb through. Most of these are class 3, you'll want a Ice Axe/crampons in winter. But, you can traverse around the south side below the cliffs to the east side where the cliffs terminate and the terrain eases.
Snowshoes in winter, Ice Axe recommended for summit block. A bike is a good idea if the gate is closed and the snow is gone.
A register was placed on the summit of "Nelson Ridge" if that helps motivate you to make that extra effort.
Just below summit
West from summit
Sitting on summit
NW from summit
SW from summit
Route looking back