"...and in fact we have been told that there is a time restriction in place meaning that they have a certain length of time to look for alternative accommodation before the property is sold, with the clear expectation that they will indeed move out."
Without seeing the actual will, this could mean one of two things.
Either that is something included in a 'Letter of Wishes' which can sometimes accompany a will. This is where the person making the will gets to say things that they would like to happen after their death. It is not legally binding. So it may just be that this is something that your parent would like to happen but it is not legally enforceable.
Alternatively, this could be written into the will, that your sibling has the right to remain living in the house for a set amount of time.
How this would happen is that after death the house is put in trust for the beneficiaries and is held for them by the named trustees (this is typically the same people who are the executors). In the meanwhile, one or more named people have the right to use the property.
This is called an 'immediate post-death interest trust'. Often, the person with the right to occupy the house will be given a lifetime right, but it can also be until a particular event happens (eg cohabitation or remarriage) or for a set amount of time.
This is typically used so that a surviving spouse is not turfed out of the family home by children who have been left a share of the house.
But it can also be used in the same way for a child (or, indeed, anyone else).
So, the important thing before you even think about any of these questions, is to find out what the will actually says.
If the will says something like:
"I give my property, 123 Letsby Ave, to my trustees to hold on trust to allow Child1 to occupy or use rent free for a period of two years"
Then Child1 has that right. There is nothing that you can do about it unless you want to challenge the validity of the will itself.
In this situation, Child1 would be referred to as the 'tenant' or 'life-tenant' and the other siblings would be referred to as the 'remaindermen'. In Scotland, they use slightly different terminology.
So, where you say "...there is a time restriction in place", everything turns on how that restriction is accomplished.
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Having said all, now to the actual questions:
Qs 1 & 2 won't apply if the house has been left in trust. However:
1 If rent is paid that will not affect CGT.
2 If you are a higher rate tax payer then, yes, you will pay tax at your highest rate.
3 With regard to CGT, if the life-tenant occupies the home as their primary residence then the house is counted as belonging to them and there will be no CGT to pay. If they move out of the house then the normal CGT rules will apply from that date (eg you will have 9 months to sell the house after Child1 moves out - unless they move into long-term care in which case you get longer to sell the house).
4 With regard to council tax, as long as Child1 is living there then there is just the normal council tax to pay (along with getting the single person discount). Once Child1 moves out of the home then the trustees become responsible for paying council tax.
If there is a second home surcharge payable then that will have to be paid as well. However, do check with the specific local authority. The reason I say this is that this surcharge usually only applies to furnished properties.
So, if the furniture is taken out of the property when Child1 moves out then different rules apply and it is treated as an empty home, but there can often also be empty home premiums as well. In the area where I live there is no empty home premium for the first 12 months that a property remains empty.