The Liverpool Overhead was the most unlikely, given that the adjacent Mersey Railway from Liverpool to Birkenhead, which was nationalised, was also electric and also fully independent. The LOR closed in 1957 mainly because significant repairs were needed to the structure which the shareholders could not afford - it basically broke even on running costs against fare income. Such would not have been an issue for the nationalised system.
The Derwent Valley, a country freight line north of York, was a curious survivor as they had given up their own locomotives long before nationalisation, and hired them and crews daily from the main system (I believe they provided their own guard). It eventually closed in sections to 1980, principally because, like the LOR, it needed major renewals which the operating income didn't justify - most of the rails had never been replaced from when it was built. Like many independent railway companies (not least the Metropolitan) it ended up making more from property development and non-rail activities than anything on the rail side.
I believe the last privately-operated tramway was the Manx Electric, which was nationalised by the IOM government in 1957, again due to running out of free money for needed repairs. It just pipped by a few months the Llandudno and Colwyn Bay tramway, which merely closed down.
The County Donegal and the Lough Swilly both remained independent because of the difficulties of nationalising something operated in two countries. Both closed down at the end of the 1950s, but whereas the CDJR continued to run a bus service for a while with vehicles hired from the Irish state bus company CIE, the Lough Swilly, still calling themselves a railway officially, ran it's own independent bus services throughout Donegal (Ireland), although headquartered in Londonderry (UK) until 2014, being made bankrupt just two years ago by the (UK) government tax authorities because they had not paid their income tax monies over. I presume that is the last UK independent railway company, although the last two extensions of the DLR in London have been built and operated by independent companies, being however operationally integrated with the main system. The Mudchute to Lewisham section is still privately owned and maintained. The same can be said about the Heathrow Express line and operation beyond Hayes.