J1 practical training/internship can be kickstarted (think the application, not entry to US soil) within 12 months of graduation, not just during studies, so depends on company that'd be willing to sponsor the visa and offer internship/placement.
More details on peculiarities on USCIS website.
They can also be tailored to one's availability/needs up to 18 months max.
The UK side of paperwork usually does/has to involve agency like USIT or other exchange agency that does the paperwork, but it should be doable with correct paperwork from US company, then doing visa interview with DS form at US embassy/consulate. No idea about changing status when in the US, regarding this paperwork.