The EHCP process takes 20 weeks if you don’t have to appeal, but sadly many parents do have to appeal, some more than once, and the waits for tribunal are very long at the moment.
Personally, if the LA refuse at any point I wouldn’t engage with mediation. You only have to consider mediation, you don’t have to actively partake in it. If the LA are going to concede they will do so regardless of whether you actively partake or just get the certificate and submit to SENDIST. Sadly, many LAs use mediation as a delaying tactic.
Parents can seek an independent assessment whether they go on to submit an EHCNA request or not. If you do go on to submit an EHCNA request you can use the independent assessment to support your request and if the LA agree to assess the report can be included in the NA.
However, personally, if you are planning to submit an EHCNA request I wouldn’t seek an independent assessment beforehand unless money is no issue. Even if you have an independent EP assessment if the LA agree to assess they will seek their own as part of the EHCNA. And if you seek an assessment now and need to appeal at any point it may well be considered out of date by the time you get to a hearing. Reports were previously considered up to date for around 2 years but over the last few years SENDIST have considered reports much younger out of date. Also, if the LA do assess you may find the LA’s EP report is adequate, but you need e.g. independent SALT and OT reports.
If the LA don’t provide the provision specified and quantified in the EHCP, following appeal or otherwise, the parents don’t need to go through the lengthy process of the LGO to enforce provision (although they may still go through that afterwards). The quicker route to enforcement is via judicial review.
You mention statemented, can I check you are in England and not NI? Because statements no longer exist anywhere but NI, and EHCPs don’t apply there.