I think its the same reason Abbey Road has no road sign... because it's a Mecca for Beatles fans... prone to either theft or graffiti.
I would like to share your optimism that the City Council have not put a street name sign up as they are concerned that it maybe pinched or graffitied. However, I think it is more likely that they have never quite got round to it. Also, the (if I may say) incredibly unimaginatively named ‘Garsington Road’ opposite Alec Issigonis Way also does not have a street name sign – I can’t believe anyone would want to nick or graffiti that!
Apparently road naming is a very contentious issue according to the Austin and Longbridge Federation. We discussed road naming at ALF's recent AGM as a considerable number of new road names are going to be needed for the redevelopment of the Longbridge site. ALF were asked to put suggestions forward which had connections to the factory which they did with many rejected for one reason or another. However, apparently mini limited editions fared well, so watch out for Rose Road, Sky Drive, Designer Way etc!