Either your spouse's passport or National Identity Document is required with the application. One or the other.
I cannot advise on Subways internship programs, however, if you wish to be sponsored by the company for a Tier 2 visa in the future I can advise that they do not currently hold a sponsor license in order to employ overseas workers.
Please see the following list for details:
The Home Office would advise that you should not attempt to travel without your original BRP card. There is a very strong possibility that you would be stopped upon re-entry without your BRP card and as such they do advise not to attempt travel without it.
Based on the Tier 4 guidance issued in January 2018, if you study a course for at least 12 months you will still be given 4 months after the course has ended.
You may be talking about the doctorate extension scheme, which is explained below:
''If you are currently following a course leading to the award of a PhD with a Tier 4 sponsor that is a UK HEI, you can apply for the Doctorate Extension Scheme to stay in the UK for 12 months after your course has ended. Once you have successfully completed your PhD, there will be fewer restrictions on the work you can do and you can use the 12 months to gain further experience in your chosen field, seek skilled work, or develop plans to set up as an entrepreneur.''
The date can be taken from the date your visa was issued on 03/05/13 and the time you spent outside the UK until your entry on 02/06/13 will count as an allowable absence.
Based on this, the earliest date that you will be able to make your application is 05/04/18.
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