Depends on whether the £2k Max budget is for just a camera body, or intended to include a lens…
The new Canon Ra is an obvious body choice, with sensor filter characteristics specifically designed for the purpose, and having good sensitivity and low noise.
Alternatively, almost any decent digital camera with the normal visible light filter removed from the sensor and either replaced with an H-alpha filter designed for photography (modifying spectral sensitivity), or, more versatile, converted to full spectrum (UV-IR) and said filter attached to lens, which can then be substituted with a visible range filter (called a hot mirror filter) enabling normal use, and enabling other applications such as infrared photography. This is a specialist modification, for which there are a number of suppliers: see the thread I also started yesterday Canon EOS-R mirrorless camera: drop-in filter adaptor & infrared photography.
One supplier in the US, Spencers Cameras, offers an optimised Astro conversion including a heatsink added to the sensor to further reduce noise, but I struggled getting responses from them when I was researching and so lost confidence in them. The other two suppliers I considered were Kolarivision and Lifepixel, using the first of these.
Apparently it is quite popular to convert a secondhand camera, not just a shiny new one, which of course can help with budget.
Then there’s lenses, or attaching to a telescope, to consider: if you don’t have a telescope, the cost of a decent one to capture anything requiring high magnification (e.g. planets) may be a lot more than the camera, but for things like the milky way or moon and, I understand, even some nebulae, standard camera lenses can be sufficient. However for fainter objects requiring long exposures motorised tracking will be needed.
When it cones to photographing through telescopes, I’ve seen some excellent pictures people have taken using quite basic cameras - e.g a webcam (i.e. cheap) - viewing through a motorised tracking telescope using multiple stacked images to effectively increase sensitivity and reduce noise.