How Do I Get A Degree In Landscape Architecture In Britain?

Basically you get onto a university course accredited by the Landscape Institute (LI, see their website).  The degree can either be first degree level (Bachelor's) or a post-grad qualification.  

To get onto a 3 year course to get a first degree in landscape design, you'd usually need to achieve at least 4 GCSEs, including in maths, science and English.  Plus at least two A-levels in relevant art or science subjects.  

To get onto a post-grad course (Master's) you would probably need a first degree in a closely related field, such as horticulture, art, botany, geography, biology.  This course may be one or two years, depending on how specialised it is (more specialised courses are only 1 year, typically).

List of Landscape Architecture courses in Britain, for undergrads and Graduate entry.

You would need to look closely at each of the courses offered by the institutions in the last list, to see exactly what their entry requirements are.

Landscape Design is a very inter-disciplinary subject; maths, science, English (needed for good communications) are all involved, with geography the glue that holds them together.