Let's see if I've understood. At your institution, the exam occurs several months after the semester ends. You would like to vacate your London apartment when classes end, and then take your exam at an Alternate Venue because keeping the apartment for the extra months would involve extra expense, which I am guessing is more expensive than housing would cost for non-disabled students, and because with your significant disability you can't just pop into London for the day to take your exam as non-disabled students can (you'd need to get established in London several days before the exam in order to stabilize your condition before taking the exam), and therefore taking the exam in London would involve significant extra expense for you. So you are requesting permission to take your exam at an alternate venue, as several classes of students, including international students, are permitted to do. Did I get that right?
I found the text of the Equality Act 2010. I confess I found it hard to read.
However, I found guidance designed to be read by education administrators, which was easier to read and generally more helpful for me. Here are some quotes. (I've done some simplifying and bolding.)
The object of the reasonable adjustments duty is to avoid as far as possible by reasonable means the disadvantage which a disabled student experiences because of their disability.
The duty requires you to take positive steps to ensure that disabled students can fully participate in the education and other benefits, facilities and services provided for students....
Where a practice places disabled students at a substantial disadvantage in accessing education, the higher education institution must take such steps as it is reasonable to take in all the circumstances to ensure the provision, criterion or practice no longer has such an effect. This might mean waiving a criterion or abandoning a practice altogether but often will involve just an extension of the flexibility and individual approach that most institutions already show to their students....
Where disabled students are placed at a substantial disadvantage by a provision, criterion or practice, the absence of an auxiliary aid or a physical feature, the institution must consider whether any reasonable adjustment can be made to overcome that disadvantage.
Institutions should not expect disabled students to suggest adjustments but where they do, institutions should consider whether they would help to overcome the disadvantage and whether they are reasonable. It is good practice for further and higher education institutions to work with students in determining what reasonable adjustments can be made....
What is the reasonable adjustments duty?
You are required to take reasonable steps to:
Avoid substantial disadvantage where a provision, criterion or practice puts disabled students at a substantial disadvantage.
Avoid substantial disadvantage, where a physical feature puts disabled persons at a substantial disadvantage; this includes removing the physical feature in question, altering it or providing a reasonable means of avoiding it....
What is a substantial disadvantage?
A disadvantage that is more than minor or trivial is called a
‘substantial disadvantage’. The level of disadvantage created by a
lack of reasonable adjustments is measured in comparison with what the
position would be if the disabled student in question did not have a
disability.
A further or higher education institution will need to take into
account a number of factors when considering what a substantial
disadvantage might be, such as:
the time and effort that might need to be expended by a disabled
student
the inconvenience, indignity or discomfort a disabled student
might suffer
the loss of opportunity or the diminished progress a
disabled student might make in comparison with his or her peers who
are not disabled.
For Example:
A sixth form college has several sites and students are required to
move between sites to attend different classes. This is likely to
place a student with mobility difficulties at a substantial
disadvantage as they may find it hard to move between sites and arrive
late for classes as a result. This is likely to be a substantial
disadvantage.
We can see at PERSONALISED ASSESSMENT ARRANGEMENTS: Alternative Venue Arrangements that KCL does allow exams to be taken at alternative venues by distance learning students, study abroad students, and international students, so perhaps you could make an argument that your institution already shows this flexibility to its students.
Another helpful document says
Factors to take into account
The following are some of the factors which might be taken into
account when considering what is reasonable:
• whether taking any particular steps would be effective in overcoming
the substantial disadvantage that disabled people face in accessing
the education or other benefit, facility or service in question
• the extent to which it is practicable for the education provider to
take such steps
• the type of education or other benefit, facility or service being
provided
• the effect of the disability on the individual
• the financial and other costs of making the adjustment
• the availability of grants, loans and other assistance to disabled
students
• the extent to which aids and services will otherwise be provided to
disabled people or students
• the resources of the education provider and the availability of
financial or other assistance
• health and safety requirements, and
• the relevant interests of other people, including other students
I also found a definition of indirect discrimination:
putting rules or arrangements in place that apply to everyone, but that put someone with a protected characteristic at an unfair disadvantage. (https://www.gov.uk/discrimination-your-rights)
Note, in the US, the government guidance in this area explains that the published examples of reasonable accommodations (I think this is the same as the UK term reasonable adjustments) are not an exhaustive list. Educational institutions are supposed to consider what the student needs, not just what has previously been offered to students with a disability. The text I found seems similar enough to the US materials that I think it would be reasonable to extrapolate this to the UK.
I think there should be an appeals procedure at each university:
If the matter cannot be resolved through your internal complaints procedure, then there are various mediation and conciliation services that might assist you in resolving the dispute.
This paragraph suggests to me that there should be an internal complaint procedure at KCL. (It parallels what is required in the US under Section 504 of the Rehabilitation Act.)
On that same page, it says
in England or Wales, a student can make a complaint that the university has discriminated against them to the Office of the Independent Adjudicator (OIA). The student will usually have to have exhausted the internal complaints procedure.
I googled for Office of the Independent Adjudicator King's College
and found three levels of appeal and deadlines for each. It looks to me like starting at level 2 might make the most sense.
If you haven't contacted these people, it might be worth a try:
Equality Advisory Support Service (EASS) Helpline
Information and advice about discrimination and human rights issues
Telephone: 0808 800 0082
Monday to Friday, 9am to 7pm
Saturday, 10am to 2pm
I hope you'll keep us posted -- by writing updates at the bottom of your question.