I took the bottom cover off and tried the second jack. Still no dial tone. Looking at how it's configured, it's hard to believe that the main jack is even wired in at all.
The incoming line (on a modern BT installation a round, black-sheathed cable which comes into the box from outside) is connected to terminals behind the upper half of the jack unit, so if you have no extension wiring it's often the case that you can't see much even after removing the lower panel:
http://www.austin-taylor.co.uk/dist/pdf/atlj02b.pdfIf you did have any extension jacks elsewhere in the house, they would be wired to the terminals on the back of that removable panel. The idea is that by unplugging the panel you isolate all of your own internal wiring from the line for testing. It serves a similar purpose to the gray demarcation boxes with disconnect points that you find on the outside of homes in the U.S.
If you have removed that lower panel, connected a phone which is known to be working to the test jack, and still can't get dialtone, then the fault lies with BT and it's up to them to fix it.
I'm guessing that a previous occupant had service disconnected some time ago before you moved in. It can sometimes happen that the records still show that the line from the house is connected all the way back to the exchange (central telephone office where all the switching equipment for your town or district is located), but that a fault has developed on the line since it was last used which hasn't been noticed. Or it can happen that while providing service for another new customer an enginner used a pair of wires at some point on the route which were formerly serving your house, knowing them to be spare at that time, but forgot to update the line records to show that the jack at your home is no longer connected to anything. So your line might run from the exchange to some distribution point (those big green cabinets on street corners) and then just stop.
Either way, if you plug a known good phone directly into the test jack and get no dialtone, it's time to call BT and report the fault (dial 151 from a working BT line, or 0800 800151 from any other phone). Depending on where the fault turns out to be, you might get a visit from a BT/OpenReach engineer.