Last time I looked, DME was not required for IFR flight in UK airways, however, ADF certainly is. Many instrument approaches do however require DME now. The marker beacons on many ILS approaches have been replaced by DME ranges.
The minimum nav equipment required for IFR flight both in and out of controlled airspace (in the UK) are listed in the ANO. These requirements in some cases are more stringent than the FAA requirements for flight in US airspace.