Yes you can start a new job while still �on the books� for another. It�s perfectly legal to have more than one employer. The only messiness will be that your first employer will not be able to issue your P45 until you have received your final pay, so you may have to go onto an emergency tax code with your new employer, but that�s easy to sort out later.
However, you should also know that your employer has to pay holiday up to you as a leaver. Basically you are entitled to the annual leave you have accrued and not yet used. The details of this should be either in your written employment terms or in a separate policy about annual leave (writing to you about it is also a legal requirement).
Let�s assume your employer is lame and only just operating within the law. According to the Working Time Regulations you are entitled to 20 days a year if you are a full time employee. So if you work half the year you are entitled to 10. Your leaving entitlement is that pro-rated amount minus any you have taken since the �holiday year� began. Not paying this up is basically withholding pay and is unlawful. Further proof of this right at:
http://www.acas.org.uk/index.aspx?articleid=80 6