Hi again. Your mother-in-law inherited the house from her late husband when he died with a base cost for Capital Gains Tax purposes of £750,000, the probate value. Her ownership period only started in September 2013 when the house passed to her.
Take a look here: http://www.hmrc.gov.uk/manuals/cgmanual/cg64950.htm
CG64950 says that where a residence is transferred between husband and wife including on death, the period of ownership of the transferee spouse (your mother-in-law) is effectively backdated to the date the property was acquired by the transferor spouse (your father in law). That would entitle your mother in law to relief from CGT for the entire period of her ownership and for the last 18 or 36 months (see below) of her ownership if she was not living in the property in that period.
However, in order to qualify for the effective backdating of your mother in law's period of ownership to her late husband's acquisition date, both spouses must have been living together and the property must have been their only or main residence before your father in law's death. As your mother in law moved out before her husbands death, this causes a problem as she was not technically living with her husband before his death. It would be very unfair for CGT to apply in this case and I would be inclined to claim the full main residence relief.
It could be argued that your in laws were "living as man and wife", albeit your mother in law was in a care home when her husband died. If you look at the following web pages, you will see that the government has made concessions whereby an individual in long term care can claim relief for the last 36 months of their ownership of a property whilst not living in it (for anybody else it is now the last months). The webpages are:
The last 18 month or 36 month reliefs have always applied to cases where the property has at some point been the owner's main home (during their ownership of it). In your mother in law's case, she didn't become the owner until some months after she had moved out. As I said above, I'd be inclined to claim full main residence relief but be ready for a challenge by HMRC. Such a challenge may never happen.
The HMRC definition of "living together" (see below) should help your claim for full relief so long as the property is sold within 36 months from May 2013.
I hope this helps but let me know if you have any further questions.