I posted the article on Sellafield in order to provide some idea of the technology. The process was introduced well after the reprocessing program. Same problem in the US. The weapons program produced tons of high level radioactive waste which accumulated in storage tanks at Savannah River and Hanford sites, and IIRC INEL. There have been major 'cleanup' program since the 1980's to deal with the waste - some of which involve vitrification. Savannah River is a relatively recent project.
With regard to commercial programs, the US suspended recycling in the late 1970's during the Carter administration. At the time, there was the West Valley Project, and limited reprocessing was performed.
A 1995 reference U.S.-German Cooperation in Elimination of Excess Weapons Plutonium (1995),
WPu Disposition Through Vitrification with HLW, C.1 TECHNOLOGY discusses the background of vitrificiation as of 1995: "Today that vitrification process is well advanced and is considered to be suitable to convert high-level waste, and in particular high-level liquid waste (HLLW) into a stabilized waste form. The technology has been developed and practiced for over 20 years. There are plants in operation worldwide, including those in Sellafield, The Hague, Mol, Marcoule, Chelyabinsk, and Tokai-Mura, and the U.S. facility at Savannah River is expected to begin operation in 1996."
Under current practice, commercial spent fuel is not reprocessed, but the goal is direct disposal in a repository (Yucca Mountain). Currently, spent fuel assemblies are stored in spent fuel pools at the reactor site, or if sufficiently old, the fuel is stored in dry storage systems - until (or rather if and whenever) the US government takes title to the fuel, transports it, and perhaps places it in the (final?) repository.