Statute lays down minimum notice periods for the termination of employment, after 17 years your employer must give you the maximum of 12 weeks notice. For you to give notice to your employers the notice period is one week as you have noted. These are the minimum periods in Law but they may be increased by agreement. The period of notice should be specified in the contract of employment but if not, there is an implied term that the contract may be terminated upon reasonable grounds (English law likes you to be reasonable). The usual definition of reasonable is Lord Devlin’s famous “man on a Clapham omnibus”, in negligence it is based on what a reasonable person might do considering all circumstances with a lower standard from a child and higher standard from a professional.
You may wish to consider:
What is normal in your trade/profession.
Your seniority, age and remuneration
How long it may take to replace you.
What is reasonable to expect from you.