Hmmm.
To benefit under the intestacy rules, a person must be
(a) a child of the deceased person ; or
(b) someone who shares a parent with the deceased person ; or
(c) someone who shares a grandparent with the deceased person or
(d) the child (or grandchild, etc) of someone who falls into categories (a), (b) or (c).
[Your mother fell into category (c), so you fall into category (d)].
Having a
great-grandparent in common would not qualify someone to inherit under the intestacy rules. So the two 'distant cousins' can't be too distant or they wouldn't qualify anyway!
Without access to the chart I can't offer any actual figures here but, for purely illustrative purposes, let's assume that there was a person, 'X', who shared exactly the same relationship to the deceased as you do (and, of course, that there's nobody else with any entitlement to inherit). Then, obviously, you'd each receive 50%. However if X had also died, leaving two children, then each of those children would share in X's entitlement, meaning that you'd get 50% and they'd each get 25%.
Note that, under the intestacy rules, the live-in partner receives no part of the estate. However under the Inheritance (Provision for Family and Dependants) Act 1975 a person who "during the whole of the period of two years ending immediately before the date when the deceased died, the person was living—
(a) in the same household as the deceased, and (b) as the husband or wife of the deceased" is entitled to make an application to a court for the terms of the intestacy rules to be varied in order for them to receive "reasonable financial provision" from the estate.
It would be entirely up to the court to determine exactly how the intestacy rules should be varied. The court could, in extreme circumstances either way, refuse to give anything to the partner or even give the whole lot to the partner.
http://www.legislation.gov.uk/ukpga/1975/63
What someone now seems to be proposing is that a deed of variation be drawn up, giving at least part of the estate to the partner. You, and the two distant cousins, would all need to agree to it. The advantage of doing so for you would be that it could avoid the risk of the court awarding everything to the partner as 'reasonable financial provision'. The advantage for the partner would be that they could be certain of receiving something, rather than having to take their chances with an application to the court.
However I'm at a loss when it comes to understanding how any 4-way split could be 'pro rata', since anything that's 'pro rata' must have a reference point from which to make the calculations and, as the amount that a court might award to the partner is unknown, I can't see how their share could be calculated by using a 'formula' (rather than just common agreement)!
Oh well, I tried!