British workplace culture is backwards as fuck
There are several issues that seem to crop up with this flexible working / wfh pattern.
Firstly, what is the culture of the organisation and even individual departments within the same organisation as to how supportive the employer is. Is there trust? is the employee in the type of role that lends itself well to FW? Is there some means by which the employer can maintain visibility that the work is still being done?.
Financial services is very much "bums on seats" in the office, even IT teams end up having to align to that inflexibility, and the "we don't want to set a wfh precedent". If telephone wok is monitored through technology (call-centre type set up with visible indicators of calls holding etc) that's only possible having the person present in the workgroup.
Other sectors are more forward thinking and even encourage remoting, such as consultancies, tech co's where people cannot do their job without significant flexibility, including wfh. It relies on a high degree of trust. Field Sales roles often need the person to remote work.
The arrangement does need to align to the person's role. If they are in a project team with internal customers who themselves aren't comfortable with interacting by Skype it means the person has to be in the office.
In short sometimes the company is back in the dark ages, but other considerations like the role itself being difficult to align to FW may be the case.
OP in your case I would be guarded about your 3 hour commute being tiring or taking too much out of your day, if you think it makes you vulnerable to them saying "that's your problem not ours". It is more likely to be approved if you turn the focus mainly onto aspects that benefit the company - such as being responsive to business need for a longer period as you can be online for more time due to not having your commute, and balancing that with days in the office for team working/interactions.
You could recommend they give you a trial period of 3 months in the new FW pattern, and keep a daily log of how much more productive you are, then proactive set a meeting to report your findings.