As you failed to name the driver, the mobile phone offence currently has no legs. They have presumably no evidence as to who was driving, and without such evidence they cannot convict anyone of the offence.
The effective charge is failing to name the driver. There is a defence if you can persuade the court (on the balance of probabilities) that you do not know who was driving, and could not with reasonable diligence determine the driver's identity. Nothing that you have told us so far suggests that you would succeed with such a defence.
Where the underlying offence is considerably less serious (in terms of points, fine or nasty endorsement codes) than the s. 172 offence, and where the OP accepts that they were actually the driver, the stock advice is to plead not guilty to both charges, but indicate in the mitigation section that they would be prepared to plead guilty to the underlying offence (e.g. speeding) if the s. 172 was dropped. However, in this case, there's little to choose between a mobile phone conviction and an s. 172 conviction.