Does anyone know where are replacement Birth Certificates obtained from ?

My wife needs one for work and only has a "smaller version" - she needs the full version

Are they obtained locally (she was born in W. Midlands) or centrally ?

You can apply online/by post/telephone centrally or you can go in person to the local office (where the birth was registered). Check the website of the General Register Office for details (search box for local offices is at the foot of the right hand index)

http://www.gro.gov.uk/gro/content/order_certificates/obtain_certificates/how--to--apply--and--pay.asp