Your best bet would be to contact the Service Personnel & Veterans Agency. They'll probably be the best people to help you. Their phone number is 0800 169 2277

I'm not sure but I don't think you can get a new copy of your 'Red Book'. But I'm sure you'll be able to get exact dates of service.

Addition: I just checked the website a little more carefully -

Quote "Access to personal data held by the MOD may be sought for many reasons, including:

proof of Service in HM Armed Forces
medical reasons
family history or research
Requests from former Service personnel

Requests from former Service personnel for his/her personal data held by the Ministry of Defence are processed in accordance with the Data Protection Act 1998; such requests are called Subject Access Requests (SAR).

A SAR must be made in writing and must be accompanied by proof of identity (e.g. a copy of a passport or recent utilities bill), and sufficient information to locate the data sought. To assist the search, your letter should include as much information as possible eg:

Service Number, Rank, Full Name, Date of Birth and period served (dates from and to).
Requests for data for ex-Army personnel must also include the Regiment or Corps.
A full copy of the record can be requested however, original documents, discharge papers/books cannot be replaced. The MOD will also provide extracts from service records. Once all the necessary information has been received, your request will be processed and you should receive a reply within 40 calendar days." Unquote