Your budget might buy you a Holland & Holland or Purdey, but it will be old and probably worn out!
It would be much better to look for a gun in good condition by another English gunmaker.
There are plenty of good examples of used, hand-made English guns on the market at the moment and £2,000 to £3,000 will buy you a sidelock ejector by a good English maker which is in sound condition.
However, if you are willing to have a non-ejector sidelock then you should get one in very good condition, as these guns are greatly undervalued at the moment.
Ejectors are not really necessary for most shooting. English boxlocks are also good value, and your budget will buy you an extremely good one. For example, the Webley 700 is a good buy as it is a reasonably recently made gun finished to high standards.
Hammerguns are also modestly priced and you could buy a nice hammergun for special shoots and have a modern over-and-under for everyday use.
Second-hand AYA sidelock ejectors are also remarkably good value. It is always best to put your money into a gun that is in good condition.