IIRC, the waterloo and city used to be connected to the mainline network via a rail vehicle lift, however it was demolished to make way for the Eurostar terminal. So all maintenance now has to be done on-site and train replacements require a big crane.
I'm pretty sure the Blackpool tramway has no connection to the national rail network. Don't know if it ever had one.
The Manchester Metrolink, Sheffield supertram and Tyne and wear Metro all have NR connections.
I'm not sure what the situation is with other tramways.
I belive the DLR is isolated, even though some of it's lines use former rail routes.
Regarding the underground, looking at the carto-metro map.
The central line looks like it has a NR connection at Ruislip Depot.
The metropolitan shares tracks with Chiltern.
The bakerloo shares the watford DC line with London overground.
The circle shares tracks with the metropolitan, district and hammersmith and city.
The picadilly shares tracks with the metropolitan and district.
The victoria appears to be connected to the picadilly at finnsbury park
The nothern appears to have a NR connection at romford.