Your scenario is a typical one. In probably 9 out of 10 cases, the answer would normally be ‘no’. The reason for this is that a legal right to light (acquired via the passage of 20 years-time and under the Prescription Act 1832) measures how much of your dining room, at a ‘working plane’ of 850mm above floor level (about desk height), can see a little bit of sky (0.2% of the dome of the sky). Only if your room can no longer see this small amount of sky over 50-55% of its area will you have a case.
In the scenario you describe, you may find you will lose ‘some’ view of the sky when looking towards your neighbours planned extension, but, most probably, your view of the sky when looking straight out of your dining room window, across you rear garden, and in the other direction, will remain unchanged. When looking in these other directions, it is likely that the view of the sky will still be visible in your room over more than half its floor area (at the working plane). Most legal right to light problems occur when a development is ‘opposite and close’ to a window.
The exception to this is when your room is very deep and/or dark to begin with. In such situations, it is possible that your room is inadequately lit even before your neighbour builds their extension. If so, any further loss of sky view may make your already bad situation worse, which could result in a legal injury and thus give you a case.