Try to get all instructions around this confirmed by email (even if that means you email them after a conversation saying "just to confirm, you advised I do x, y, z. I'm going to check with HR how to do this as this is all new to me, then will proceed. Thanks for your help.").
Also, make your own notes on all the conversations re being "managed out" (dates, times, location, witnesses, etc).
You've got a great opportunity to protect yourself here on the basis you're providing temporary cover in this role and haven't been trained on all the procedures. This makes it perfectly reasonable to be checking with HR every step of the way (which a manager in this position should be doing anyway if they're unsure) and to get it crystal clear as to the grounds you're using for each step of the formal process, both so that you can communicate these to the employee and so you're clear yourself on the law and the procedure.
Also, have you checked the job description for your new role to work out what your role/responsibility is in respect of formal HR processes? It might not say much, but there might be something useful in there, depending on how detailed it is (it might say "with the support of your line manager, HR, etc" which might be helpful to justify why they need to be involved too).
I also wonder whether they are taking advantage of the fact you're new to this role, and how much support from HR the old manager would have had on this. I can't imagine you'd be left to do this without HR support in a big corporate, but that may be exactly why they've picked on you to do this. Part of being in senior management is pushing back and if they are going to give you the responsibility to carry out this task, you also need to have the freedom to make your own decisions and use your own judgement. Either they don't want your expertise and judgement, in which case, they can deal with unlawfully "managing him out" themselves, or they can give you the responsibility and autonomy to do it properly, involving HR and lawful procedures.
As a PP says, you will end up giving evidence at a tribunal if this gets handled wrongly, so it's definitely worth covering yourself and making sure you don't do anything you disagree with unless someone superior is prepared to authorise you to do it in writing.
I would also be mindful of your own employment protections in all of this - if you are feeling coerced and that your own future at the company would be in jeopardy if you fail to follow unlawful procedures, it's important you get advice (try ACAS) before doing anything and before approaching HR.
Also have a look at your company's policies (both disciplinary and grievance and whistleblowing) to familiarise yourself with your company's official position on all of this stuff.
Sometimes just saying "that goes against our disciplinary and grievance policy, so I'll need to check with HR" is enough to alert colleagues to the fact you're not going to be used as a patsy and are going to do things by the book. Once they know that, they may well back off (this has worked for me before - colleagues soon learn who will help with their dodgy dealings and who not to bother asking).