Looking at the routes, seems like the best place to meet up would be the intersection of Saltillo Rd (HWY 2) and HWY 77 in Southwest Lincoln. I would expect the Omaha crew would come in on 77 and coming from Elmwood would be Saltillo Rd (HWY 2).
There is a Shell station on the Northeast corner of that main intersection:
Shell · 1200 Saltillo Rd, Roca, NE 68430 - https://maps.app.goo.gl/81gprWjin2nhxHrs8
It's about 6 hours from there to Sharon Springs. What time do you want to get to Sharon Springs?