No it shouldn’t affect your chances in theory. No one cares where your degree was from or they would say so in the entry criteria. And distance learning doesn’t diminish the value of your degree either.
However, your degree is not just a ticket of entry into the running for CP. Your degree is your opportunity to really explore your subject and to make connections with working academics, with clinical experts etc who might be able to shape your research interests, put you in the right direction for further opportunities etc. Distance learning arguably provides more limited opportunities to do this, especially in a conversion degree which is only a year long. On distance learning courses, you need to make sure that you’re actively seeking opportunities outside your degree or possibly within your degree to really bolster your CV and your profile.
I’d also encourage you really think carefully about whether or not this is the best use of your postgraduate funding. What are your motives for wanting to become a CP? How well do you understand the role? Have you considered any alternative, related careers? Are you interested in research? CPs study for a clinical doctorate and they need good research skills and need to be motivated by research. If research is something you think you’d only ever do for the sake of becoming a CP and you can only picture doing the bare minimum that you have to, this may not be for you.
If your main interest is counselling or talking therapy, please remember that you don’t need to become a CP for this and there are actually many career paths that are far more accessible that would enable you to do this. Mental health nursing is one of them and other allied health professions may also provide these opportunities. You could also do a master’s in CBT or other therapies that would allow you to practise as a counsellor. I do encourage you to consider these routes as well.