Just be aware that Subject Access Requests are not handled under the Freedom Of Information Act, they are handled under the Data Protection Act. Completely different kettle of fish.
As far as I know (and that is not a lot), submitting a SAR will not expedite your application in anyway. SAR is a request for the Home Office (and other government institutions) to give you a copy of the data it has on you. The Home Office (and other government departments) do not give an explanation of the data, it is literally just a data dump, a printout of all the information it has on you (including scanned copies of applications, landing cards, etc). And be aware that due to the DPA, records are deleted from time to time. So, it is possible that a specific record from a few years ago may have been deleted and hence no longer accessible via SAR.
The way that SARs come in handy for immigration applicants is that if the application has been processed, any comments made by caseworkers on the file will also be accessible (suitably redacted). That can help you shape any future application.
What it does not do is push your pending applications to the front of the queue. If it has been opened, it should be included in the SAR report, but that does not jump it to the front of the queue.
More information on SARs on the Information Commissioner's website.
Freedom Of Information requests are for a different purpose altogether. It is to ask the government for any written records on a particular topic on which they hold such records. It is typically wider in nature. So, assuming that the records exist, you can use FOI to ask the government how many naturalisation applications have not been decided within six months.
The difficulty is that FOI applies only to written records that exist already. Therefore, the government can avoid publication simply by not having any written records of that specific nature. There is no obligation for the government to compute how many naturalisation applications have exceeded six months. If it has not been computed and written down already, the FOI request can be declined, because there is no existing written record.
Also see this
post about FOI.
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.