Thanks for your question. I will try to help.
I’m afraid that there is not much that you can do to speed up the process if the home office has refused to expedite the hearing. They are correct in that there are many people in this situation and they cannot really give preferential treatment.
If the test results are fine then the best thing that you can do is to instruct a solicitor as soon as possible to act on your behalf now that the old solicitor has ceased acting.
You should instruct a UK based immigration solicitor from a medium sized firm close to you. You can find immigration solicitor using the law society find a solicitor website here:
They will arrange for a barrister to be instruct on your behalf in the matter so that they will act at the actual hearing.
They may even write to the home office before the hearing starts with a copy of the test certificate and requesting they withdraw from the appeal, since the overstay is not going to enough of a reason for them to reject the appeal on its own unless there was any deception (which I assume there was not).
The only alternative would be for you to instruct a solicitor, withdraw the appeal and then submit a new application with the test certificate and correcting any other omissions from the original application. If the only reason for the refusal was the overstay other than the test certificate then the solicitor should be able to submit a covering letter arguing that the overstay should be disregarded. This may result in him getting his visa quicker than at tribunal, but obviously if the application is rejected then you would have to appeal and would be placed at the back of the queue again.
I really am sorry.
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