A clearly defined contract is preferable, however it is most certainly not a requirement should you wish to pursue legal action. The OP has stated that he has hundreds of emails, so there is definitely a 'paper trail' which can be used to prove what was agreed between him and the client.
This seems pretty clear cut and if he has an email or emails that prove that the client instructed him to build a web site for £xxxx, then he most certainly should not just walk away from this, there's still a good chance he can recoup some or all of the money owed.
OP - it's time to send a letter to your client, sent via registered post. This should be formal, clear and concise, written with no emotion - not 'friendly', but not angry - be strong and firm. Lay out what is owed to you (include an invoice), payment terms, a reminder of what work you performed (include a copy of the relevant email/s) and timeline for payment (e.g. 30 days NET) and most definitely lay out exactly what you intend to do should no payment be forthcoming - i.e. legal action.
This may be all you need to prompt the client to cough up what he owes you, or certainly mark the beginnings of some kind of negotiation. If he ignores you, follow through with another letter, i.e. a final warning before legal action. Consider having a lawyer draft up this letter.
Many of these kinds of issues normally get resolved without the need for full legal action, but remember that small claims court might also be an option.