Part-time PhD workload compared to part-time undergrad workload
A bit late with this answer but hopefully will be helpful. You ask;
Should I expect my personal time to diminish significantly when I continue on to postgraduate study, compared to life as a part-time undergrad student?
Judging by my personal experience the short answer is yes.
Here's a bit more detail though. I am studying in Ireland, so not to dissimilar to the UK system. I work full-time and also did an undergraduate degree, part-time, before progressing to post-graduate study. My degree and post-grad are not related to my work but something I have a lot of passion for. I found with my undergraduate degree I had a lot more free time. Lectures were 3 nights a week for year 1-2 and 2 nights a week in years 3-4. Speaking for myself on top of lecture nights I found I could usually get away with doing a bit of extra work at the weekend.
Two years ago I started a History Masters(which I have just transferred to a PhD) and found that almost all my free time is now used with doing my PhD work. Saying that it's important to have other activities outside of your full-time work and study. I follow my local football team so usual have a evening off each week.
As you mention you have not decided on location so I'd like to add something about my experience. My workplace is about a 5 minute walk from my University. I think this makes all the difference to me. I can leave work at 5pm, have a coffee and be working on my PhD by 5.30. I also have my own dedicated desk space there which is a benefit with regard to having a space to leave books etc. Together with working in a family business this also gives me the flexibility to be able to arrange meetings with my Supervisor at working hours times and to pop over to the university during the day if I need to. So if possible I think it would be good to consider location of your institution in your decision making.
Just to sum up my weekly time spent on my PhD (and this can always vary depending on other commitments.) About 4 nights a week 6pm to about 10.30pm, Saturday 11am to about 10pm and Sunday about 1pm to 7.30pm.
We have the OU in Ireland too. The university I go to is as StrongBad describes in their answer a bricks and mortar university. In my own situation I have found every department that I have dealt with to be very understanding to the part-time nature of some of its students and to be honest the only situation I can think of I have been disadvantaged by not been a full time student is the opening hours of some of the offices. That said most functions of the college can be done online now or I can email the respective office.
Note: I did a Masters full time and half of my PhD. For the last year of my PhD I got a job as a professor, so the work I do for my thesis overlaps with my research work.
I think you'll find that the course load won't be much different from your BSc. For me, it was actually less work.
However, you will spend a huge amount of time on your thesis, particularly if you go for a PhD. The problem is that you will be doing research, not just the class assignments. And research is inevitably uncertain. You don't know the right answer (or if there is one) going in. So you should be prepared to spend a lot of time trying to solve problems and come up with explanations for your results. Insight and invocation usually require thinking about something over a long period of time. If you're constantly distracted from your research, it can be hard to come up with solutions to the (many) problems you'll encounter.
Since you mentioned your doing computer science, you probably won't have to spend time "in the lab", which is a huge time sink (my girlfriend is a biologist, she spends 6-10h a day doing experiments). My PhD topic is highly computational (I spend most of my time programming) so I do things from home or anywhere else I can take a laptop to. I imagine you'll have the same luxury. Even so, I'm still having trouble balancing my teaching load with research. It's very hard to have insights when I can only think about my topic for 2-4h a week, or not at all for months at a time.
In short, the problem is your thesis. It's possible if you don't depend on spending hours a day collecting data. It really helps if your thesis topic overlaps with your job. Be prepared to dedicate many hours to thinking about your research topic. As with everything, it depends a lot on how you function personally.
I think there are two aspects of your question. The Open University (OU) and its faculty are very familiar with the difficulties and challenges that part time students face. My guess is that the additional workload going from a full-time-working part-time OU undergraduate student to a full-time-working part-time OU PhD student is substantial, the OU has the support mechanisms in place for you to succeed.
As a full-time-working part-time student moving from the OU to a brick and mortar school, will likely be a shock even if you stayed as an undergraduate. The OU is really designed to do what you are doing and while part time studies can be done at a traditional universities, it is not what they are good at. The increase in difficulty and time commitment at a traditional university for full-time students as they move from be an undergraduate to a PhD student is substantial. Coupling that with the loss of the OU support network, is going to make it even more difficult.
Should I expect my personal time to diminish significantly when I continue on to postgraduate study, compared to life as a part-time undergrad student?
Your personal time does not need to diminish, but the "attacks" on your personal time are going to increase greatly. You are going to need to have a good working relationship with your advisor to define boundaries that are mutually agreeable.