What formalities do we need in order to effect the buyout? Is it dependent on legal separation / divorce?
It depends how much you agree/on how good terms you are.
First, if she does not have savings to buy you out, then she would need a mortgage on the house or an equity release with your permission as current joint owner. There would likely be a Land Registry TR1 form for you to sign to transfer ownership to her and to be executed in exchange for payment.
I'd say you should have conveyancing solicitors for each of you taking care of that. My ex and I did it with one set, and while it worked out in the end, in hindsight I cannot generally recommend it.
As for it being dependent on legal separation/divorce, on the ownership side, until you have a clean break, in principle one part can get a share of the other's property through the courts.
On the purely financial side, if you need a mortgage for a few years before you retire, then some lenders will insist on shared ownership - both of you on the deeds if you are married. That means you will both need to be checked for affordability immediately and you will need a second check if/when you take over the mortgage for yourself.
Other lenders will not insist on this. So if you need a mortgage to get your own place, then you will have more choice (and probably less hassle later) if you are divorced, but it is not impossible.
Is any stamp duty due on this sale?
Beware that the legislation has just changed, but generally speaking: Not if you are buying another place to live in yourself and you sell the existing place at the same time or less than 3 years before. (On that note, I believe you are exempt from capital gains tax for 18 months after moving out.)
If you sell afterwards, then you may have to pay 3% of the purchase price of the new place upfront and then reclaim later (but not later than 3 years after).
There are many variations, particularly if you own another place; the one I will highlight is if you buy a place before you are separated beyond reasonable doubt, you may have to pay 3% SDLT on the whole price upfront as you and your wife would then own two properties, at least temporarily.
Proving separation earlier than a Decree Nisi is not something I have looked into, but I know that there is something called a "Deed of Separation", which can be drafted and signed.
This is only viable for me if it’s a clean break financially and she has no further claim on my pension. Is there any reason that can’t be done?
As your children are effectively able to take care of themselves, in principle there isn't if you agree on everything.
If you do not agree on things, then the final settlement will likely need to be "fair" in court terms and again, legal advice is strongly recommended apart from reading posts here. You can perhaps ask the solicitors drafting the clean break for a start if you are close to that point.
By fair I mean that it is balanced enough that the courts will approve it.
For short marriages without children there is some latitude, it may be considered fair that each take with them what they brought into the marriage.
For longer marriages the split will have to be more equal, think of a D81 form where you declare your savings, pensions, debts, valuables/property and net income. If you both earn approx. the same and intend to split money, debts, home etc. equally, then that is probably fair.
On the other hand, if you or your wife have spent time taking care of children rather than working on a career, then that may be ascribed a value by the courts and "fair" may mean that person gets a bigger share of home, savings etc., regardless of the reasons for divorce.
But if you have both taken legal advice, both freely agree on something and have signed a consent order to that effect, then as long as the children and perhaps other dependents are provided for, it is unlikely the courts would interfere and object.
Also note that the courts will not seal a consent order before your Decree Nisi, and if you want a "fault free" divorce or the closest we have in England/Wales, then you need 2 years of separation plus the time it takes to apply and get the Decree Nisi before the consent order can be sealed.
Until then, it may not be practical to try to enforce an agreement as the courts could override it anyway when the divorce is finalised.
A consideration is death benefit from my pensions.
How would this be affected by separation / divorce?
As other posters mentioned, it is best to take some legal advice in case of doubts between you and your ex, but as a pointer, a normal pension is not shared/transferable after divorce (Decree Absolute).
However your ex may be able to claim parts of them through courts (again, you and her may know she will not do that - I'm just saying that it is legally possible).
Clean breaks/consent orders typically include clauses that prevent claims on pensions for that reason. You can decide not to include them, but then it is not a completely clean break, so to speak.
Finally, either of you can get the Decree Absolute before the clean break is approved by the courts as long as the respective required times have passed since the Decree Nisi, on the other hand, the Decree Absolute may work as a "carrot" to get the clean break through.