Whilst the employer should deal with his without unreasonable delay, I am not aware as to why the actual delay has occurred and there could be reasons for that. For example, witnesses may be unavailable, those who would be expected to deal with this may be away or otherwise engaged, etc. So there could be a reason, or more, which can explain the delay, although I of course understand it is not ideal just to be left hanging.
As you cannot force the employer to expedite this process, you could consider making your own complaint against the employee who made the false allegations. You have the right to raise a grievance against them because they have acted maliciously and made a false complaint against you so you can complain to the employer and let them deal with this. In the meantime they should hopefully resolve your investigation. If necessary you may also include this delay as part of your complaint.
The other option, which would be a drastic one, I to resign and claim constructive dismissal. However, it would be too early for me to recommend that and it should only be used as a last resort so go down the grievance route if needed first, before you consider the constructive dismissal route.
I trust this has answered your query. I would be grateful if you could please take a second to leave a positive rating (selecting 3, 4 or 5 starts at the top of the page). If for any reason you are unhappy with my response or if you need me to clarify anything before you go - please get back to me on here and I will assist further as best as I can. Thank you