Thank you for your reply.
The agreement among the Trust, Institute B and my supervisor was to accept the fact that we need to be transferred as part of the group since I already started my work contract at Institute A. Staying with Institute A is not an option if my supervisor is moving. Since my position is funded from grant X (by the Trust) which is now moving from Institute A to Institute B, does this make a transfer possible?
If a new visa is to be applied for. How could my supervisor coordinate the resignation from Institute A, the advertisement of job at Institute B and the application of the new visa (time wise)? The idea is to start at Institute B on the 1st of June, but we are flexible.
The point obviously here is I want to avoid having any gap which may put me at risk of being asked to leave the country.
Currently, my dependents are living in EU (student visa), but as I mentioned they already got the Tier-2 General dependent visa (UK). Is there any advantage of asking them to move at once to the UK before a new application is started (time wise, visa fee, complications...)?
If my presence is not required for the time being, either in Institute A or Institute B, and I can work from home. Is it possible to be in my home in EU (rather than in the UK) during this process?