Everyone would die, and noone would build anything.
Without supplies and materiel, I would have expected a massive death rate in Sol after Lucy popped.
On the ability to build fleets however, I'm not sure. Sol was still very much the center of the GTA, and even after 30 years the GTVA fleet really wasn't that big; I'd estimate no more than several hundred ships. I didn't really get the impression it was as large as the GTA fleet at the end of the Great War. So whatever the bottleneck is for construction in FS, it might not apply to Sol, since they do have SOME resources and most of the shipyards. But the economy would require massive restructuring, and since they may not have immediately noticed the node not working, poor management and a desire to keep the situation from the public could result in large shortages. I figure a Stalinesque approach to economic change, with billions of deaths.
Lucys head would be totally useless; until I rewatched the end cinema from FS I thought I remembered it being largely intact, but it's totally blown to ****. Its as usable as the Galatea, ie Not At All.
And anyway, Sol is likely to be immediately torn by internal division. While Sol:A History is exaggerated for dramatic purposes, its almost certain that the GTA government would fall; frankly I'd expect rioting in the streets and massive disruption. Once things calmed down they'd have a lot of catching up to do, and really, with absolutely no concievable threat, why would you concentrate on military technology beyond Keeping The Workers Down and Looking Hardcore On Television?