Did they say no to your part time request or did they say yes as long as we can find a job share and then you believe they didn't try hard enough to recruit someone? Not quite the same thing. Anyway, that's over now and not relevant to the problem you have at the moment.
In terms of the role you did accept, as squiffy says, the fact that the part time role they were able to offer you was on a lower scale and less money isn't necessarily a problem, as long as the scale and money reflect what the job is. You were under no obligation to accept it, but you did and again, that's all over now.
IN terms of the new position offered to your colleague, well if her role didn't exist anymore and the alternative was I imagine making her redundant, it sounds absolutely right that it should be offered to her rather than you, as you were safe in a job.
So as I see it, on reading your post, the only actual problem here is that you believe the job title you have doesn't reflect the actual work you are doing. Do you believe the money you are on doesn't reflect the work you are doing either?
Whether it's just the job title or the money as well, it's fine to be fed up with that, and to try to do something about it. Assuming their promise of looking into it in April isn't soon enough for you, put together a case for a new job title and salary (if appropriate). Outline the duties you are taking on, give examples of your excellent performance and of you going over and above the call of duty in the last few months. Give comparisons with similar roles in similar industries if possible.
In other words put something together that doesn't sound like a whinge, but sounds like a well-thought out, rational, logical and business-like proposal for your role going forward. If you have an appraisal coming up, that would be a good opportunity to do it, otherwise book a meeting in with your line manager to discuss it.