Aimsmum you don't have any specific rights to extra concessions being made while you are off sick, however there are two points I would make. I think your injury may well qualify as a disability under the Disability Discrimination Act - see here for information about deciding whether you are 'disabled' or not. If you are, you are entitled to reasonable adjustment to enable you to do your job.
Leaving that aside I would do the following.
You still have time before the new deadline. Contact someone more senior than whoever you have been speaking to and explain the situation, say you are unable to complete the report as you are off sick and do not have access to what you need and have been told an extension to allow you to do this has been denied, could this decision be reviewed. Do this by phone I would say rather than email.
Assuming you either can't do this or don't have success with this, I'd do the following -
Don't do the report. You have advised someone that you are unable to do it without access to your PC, so stick to that and don't do it. Instead write a letter to whoever would make the decision ultimately about your job, whether it's someone senior in HR, or another senior manager, or whoever. Say something like -
As you may be aware, I am currently signed off sick following an injury. I have been signed off for 8-10 weeks however I hope to be back at work sooner. I advised [name of person] on [date] that I was unable to complete the report necessary for my appeal at present as I require access to my PC and obviously I am currently unable to come into work at this time. I requested that in this exceptional circumstance the deadline be extended to appropriate date in order to allow me to provide you with all the appropriate information for your decision about my appeal. This request was denied.
I hope you would not feel it appropriate to make such an important decision about my future without all the necessary information and without giving me the opportunity to make my case for appealing the existing decision. This appeal process has now been going on for a year, and it is difficult to see that a request to extend the deadline by a few weeks in order to enable me to complete the necessary report is an unreasonable one in that context.
I believe that my injury qualifies me to be entitled to reasonable adjustment under the Disability Discrimination Act and would propose that a short extension to enable me fair access to information which is readily available to others in the same position is a reasonable adjustment which can easily be made.
I look forward to hearing from you.
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Send it in letter form to whoever the senior person making the decisions is, and by email as well, and copy it to your contact in HR, your own line manager and anyone else who is relevant.
Then just sit tight and see what response you get. If the decision is made without your report following a letter like that being submitted, you should put in a grievance, which I think would be reasonable whether or not you qualify under the DDA.
That's what I'd do. I think whether or not you qualify under the DDA this is not fair or reasonable on their part, and a failure to enable you access would merit a grievance, but adding in the DDA bit if you feel you qualify would mean there is also a legal basis to your request and might make them more likely to give you the extension with less trouble.