Driver who hit me not admitting fault

Based in UK

Long story short, driver hit my car whilst I was stationary.

Initially he wanted to settle privately and for me to not go through insurance, so have a series of WhatsApp messages apologising and saying he will pay as it was his fault. However when I got a quote for repairs he asked me wait nearly a month for the money, so I decided to just go through insurance to get the ball rolling asap.

I have uploaded all these messages to my insurer but they haven’t managed to get the money back from his insurer as they have not admitted fault.

I’ve been told that this will be passed onto solicitors as a result of this.

Is the evidence of him messaging me admitting fault enough? Surely solicitors will see that and not bother taking it to court?