Professional advice is the key, we can all provide opinion, but that is what most of us are able to do, and little more than that opinion.
I would imagine that your sister in laws survival mode has kicked in therefore causing her to suggest this offer. I would imagine that deep down she knows its good for her.
Contact citizens advice, they can be known to provide excellent advice FOC.
If it was me, I would just say, that you wish to run a business alone. Whether that be the current company trading name or a new trading name.
Make sure your accounts are up to date to enable a fair assessment of past and CURRENT trading status to be considered should it go that way.
I would not wish to go into business in this way from here on in. A working partnership is fine, as per that with your brother, but this one would be a noose around my neck, personally I think.
Add up the value of your equipment, you could either be generous and look at replacement cost, or alternatively look at the value within your accounts to assess the depreciated cost. As an accountant I would envisage the Sister in law would take the depreciated cost, as it will be much less.
Then consider whether the company name, phone number etc is worth anything or is it your face that is the value of customer loyalty etc.
Look at any contracts that are written up, or are they all as and when verbal agreements, that could be taken away at any minute. If the latter, then maybe they hold little value upon a commercial business valuation assessment being carried out by a specialist company.
If this was me, and this is a opinion of what I would do, not what you should deem fit. I would be looking to establish myself as a sole trader in arboriculture. Either by paying off the sister in law and getting it written up and acknowledged in writing professionally. Or set up again. Im guessing you have contact names and numbers of customers to hand at your home address, and they are not all at your brothers house. This may be seen as niggly, but you have said you have kids to, and maybe your 'survival mode' needs to kick in a tiny bit more for you to see what needs to happen for the benefit of your families future initially.
I would be open about how you feel, and simply say you do not wish to look into the possibility of the sister in law being involved in the firm, and wish to 'dissolve' the company, or buy her out as deemed appropriate by uk law.
Condolences on your loss. Good luck with it and hope work picks up soon.
P.s. Just googled business partnership buyouts, Loads on there relating to it. Have butchers.