Each route will start as soon as you cross the state line. Each State should have a cooperative agreement with Virginia before their respective sections were started
North Carolina has already said they will not be upgrading US-220 to interstate standards north of NC-68 until Virginia begins their portion near Martinsville, such as the Martinsville Southern Connector project extending the US-220 freeway to the state line.
NC-68 will likely be the northern terminus of I-73 for quite awhile.
Likewise, they will not construct I-73 south of the Rockingham Bypass until South Carolina begins their portion. The under construction Rockingham Bypass is likely the last I-73 segment to be completed for quite awhile as it will finish the interstate throughout the state with the exception of those connecting pieces mentioned above.
As for I-87, the state does plan to complete the interstate to the state line, potentially beginning around 2027. What Virginia does with their segment... who knows. The 12 mile US-17 segment in Virginia is currently under study and is one of the candidate projects for the 2045 LRTP in Hampton Roads... but that’s about it. Until the segment between the Elizabeth City Bypass and the state line is built, there will probably be nothing done in Virginia.