Sorry this is going to be a bit disjointed, I'm jotting thoughts down, but hopefully will make sense.
Employment tribunal can force an employer to hear the request again, this time following the correct procedure, but they do not have any power to say whether a request is reasonable or not. So from that point of view it's not going to help get the decision changed, unless they are basing the decision on incorrect facts and therefore need to be forced to reconsider it.
It doesn't sound as though that's your problem.
I very much doubt you could prove there had been a blanket policy decision, so I'd forget that as an option. In any case, if it's all very similar jobs, for example, consistency in terms of what is and isn't possible wouldn't be surprising.
Complaining about who is making the decisions isn't a way to go either. You have no proof that the more senior managers you say are making the decisions aren't being fully informed by the relevant more junior managers, which would be what they say, and would be fair enough as well.
You mention an unspoken policy of not allowing part timers, but that's not the case is it? They are allowing part timers, just not 3 days?
The bottom line is the employer must consider a request, they must follow the correct policy, and they must give one or more of eight set reasons for refusing, and explain how that reason applies in each individual case.
Obviously I have no idea whether your wife's job could be done in 3 days or not, so can't advise whether their decision was reasonable. I would always advise someone to explain clearly how this would work in their application, and the advantages for the employer in agreeing. With a request to work 3 days instead of 5, that's a fair chunk of the working week gone. Was your wife making a realistic and reasonable proposal as to what would happen to the other 2 days worth of work? Was she proposing a job share or something?
In terms of returning to work, your wife's boss shouldn't be pressurising her for a decision. An employer must assume a woman will take the full year maternity leave unless and until they receive written notification that the woman intends to return earlier.
I don't think they can make her submit another request for a solution proposed by them, no. Has she been in touch with HR regarding this? Particularly as this solution has been mentioned in writing.
Next step is to think very carefully about whether the reasons given for the refusal of 3 days are reasonable, without allowing any anger or frustration to cloud thoughts on this. One or more of the following reasons must be given, with an explanation of how that reason applies.
Burden of additional costs
Detrimental effect on ability to meet customer demand
Inability to reorganise work among existing staff
Inability to recruit additional staff
Detrimental impact on quality
Detrimental impact on performance
Insufficiency of work during periods the employee proposes to work
Planned structural changes
As you can see, the reasons allowed are fairly general and give an employer a fair amount of leeway to refuse a request if they want to.
If your wife honestly feels that the grounds given for the refusal doesn't meet the requirements, she could consider putting in a grievance.
Another next step is to sort this issue out about another request for 4 days. It's already been mentioned as a solution, and has been accepted by your wife. I think her boss is just being obstructive, I doubt that advice has come from HR, so she should speak to HR about that and get that resolved.
It's very frustrating, and I can understand the feeling of more personal betrayal as well given the circumstances, but it's important to look at the bare facts in the cold light of day, and consider them along with considering what the best possible outcome is for your wife, and how to achieve it. If she wants to stay in her job she will need to continue working with this person, and resolving this as soon as possible will prevent that becoming even more difficult.
I hope that helps a bit.