In the UK think of:
UK Class D as US Class C
UK Class A as US class B
UK Class B as US Class A
(gave the teeside controller a snigger when I read back "cleared into Teeside Class B", to which he responded something like "I don't think you'll be entering our Class B today, however you're welcome into our Class D"
)
Each UK Class D airport will have an approach control facility, but there is no "common" approach control, like SOCAL or NORCAL in California. For the purposes of Airways IFR flight in the UK this would be London Control or Scottish Control (Equivalent of a Centre in the US). The Channel Islands are a bit different. Things there are more like you find in the US, where Jersey Zone controls all the incoming flights and then may hand them off to Gurnsey Approach who then hand you off to the tower of the Island , or straight to Jersey tower.
Class D in the UK is more or less exactly similar to Class C in the US. You need an ATC clearance to enter and an ATC clearance to depart ("Cleared to depart to the NW, not above 2000' "), you will normally be handed from the tower to the departure controller on take off (Often called "RADAR".....G-YZ contact XXXX Radar now on 123.45 "). OK strictly speaking in the US you do not need an ATC clearance, just two way comms, but its very similar.
Class D in the states normally does have radar, however it is provided by the approach control facility, and just piped into the airport tower. US D is equivalent to our ATZ's in the UK,....in fact if you treat airfields in the UK like you would a Class D airfield in the US you can't go wrong. Establish comms at an ATC field before entering their A(ir)T(raffic)Z(one), try and establish comms at an uncontrolled "Radio" and if no joy, just broadcast your position and intentions.
The differences are not that great, the phraseology in the UK is slightly different, but more precise than the US which if anything makes it easier to get into.
There are other differences though, which FFF has pointed out, requesting "left Close Traffic" will get a big "ehhhh, say again" in the UK
Hope this helps a bit,
EA