I studied in that department and under one of the admissions tutors for that course, that I didn't study that specific MPhil.
Whilst I can't comment on the exact requirements without knowing the content of your course this is taken from the website:
The MPhil in Development Studies is largely for students with a background in one or more of the social sciences, although many students have had previous degrees in disciplines ranging from the humanities to the natural sciences. The academic backgrounds of the current intake, for example, include American Studies, Anthropology, Biological Sciences, Business Economics, Business Management, Computer Science, Development Studies, Economics, Economics and Politics, Engineering, English, Finance, History, Information Studies, International Relations, Journalism, Operations Research, Philosophy, Political Science, Public Administration, Public and International Affairs, Quantitative Economics, Social Studies, Social Work, Sociology, Spanish and German, and Urban and Regional Studies. The normal requirement for admission is the equivalent of at least a high 2.1 degree in the British university system (minimum CPGA 3.7 out of 4.0 in the US system). Account may also be taken of other relevant qualifications and experience.
I can't comment on the written work but if it's of a high standard then will help. You would need to communicate effectively in your statement why you want to study this course and how it will benefit you in the long term and perhaps propose a suitable research topic in line with the interests of the members of faculty. The latter will be particularly important given your somewhat unconventional background in applying for this course and I would recommend visiting the faculty website, reading about research interests and perhaps emailing to clarify your eligibility and chances of admission.