Sorry Chris but I wouldn't ask our political leaders the time. Let's figure this out for ourselves.
When you say ". . to what extent do I have responsibility and effective control over the home working environment . . " are you speaking as an employer or a safety adviser? It is of course the employer who has s2 responsibility, but only in so far as reasonably practicable, given the actual degree of control. Which in the case of private homes, is pretty limited.
Background: I worked from home for my employer for 5 years, most employees were home based. They had a comprehensive guidance and policy pack, plus induction training (although back then it was on the head office site). Some loved it (me) but others found it difficult and expensive as a long term arrangement, as they realistically were providing a chunk of real estate to the company for very little real benefit. Others just found it soul destroying, missing the connections and interaction of the office. As a new worker, it took a couple of years to get to know others useful to my network, via phone calls, emails, occasional meetings, and a 2-day conference every few months. Before Zoom obviously. I learnt a lot during that time around home working, and have been called upon since to develop policy for homeworking in a couple of other organisations.
Lets go through this: The employer will be providing (let's assume) at least a laptop and some IT access. Of course the employer is responsible for the inherent safety of equipment, and also that workers can achieve a reasonably healthy and comfortable working posture / position, without being able to dictate exactly how. A key step is to educate managers, who may also be home working, on how to manage a dispersed out-of-sight team giving up a lot of autonomy and moving away from a single standard.
Practical list: With every laptop, provide as routine, an additional keyboard and mouse or other inputting device PLUS a laptop riser. Will also need a multigang socket / extension, and a work mobile phone charger. So are you going to add a smoke alarm and a socket tester or RCD?
Each worker needs to have a discussion (call it an assessment if you must) regarding their home situation for work, whether there are specific issues or barriers to be taken into account, and what - if any - additional equipment the worker suggests, either then or in the future. If the worker needs and asks for a chair, let them choose, as they have to have it in their home, so it needs to suit them. An office chair is only any good if they already have a desk or table with space which they can use. If they live in a shared house even that could be a problem. Not everyone has a spare room or garden chalet they can dedicate to work. So the employer is not in control of anything very much, let alone staff tripping on the frayed carpet whilst carrying the laptop.
Safety advisers can produce guidance for managers to give to home workers on setting up a comfortable work position, good practice around electrical safety, reporting problems or concerns, and advice on a good work life balance. There is plenty out there on adapting laptop workstations and dealing with practical aspects such as making a dining chair more supportive. Their role will become a mixture of supporting managers (as usual) and possibly helping individuals more than would be the case in the standard office. Where workers ask for a bit of kit, chair, folding table, etc. have a way of letting them choose and / or source.
The future looks like more companies will stick to having home working, with those for whom it does not really suit to be back in the office but with more space. We have to keep in mind that office work is pretty low risk, even with toy-strewn floors, sleeping dogs and juggling a laptop on the back of the sofa.
HOT NEWS: There is a virtual Working From Home Show on July 30th/31st with a H&S panel at 1.45pm on July 31st. See https://workingfromhomeshow.vfairs.com/ Warning: I am a panellist. I did provide a list of resources but not sure how if at all that will be offered. You're welcome to PM me for a copy direct email.