It's a long way Mummamoo, there are very few allergists and waiting lists, particularly if you live out of London are almost eternal (DS reacted to peanuts in March 2005, was refered for testing one week later, and finally was tested 3 weeks ago), in the mean time we tried to go the private route but couldn't find ANYBODY who could see him in less than 3 years, we ended up having him tested abroad. The skin prick test his doctor sweared was the only reliable test (it's all matter of preference TBH) only confirmed what the other tests had already told us.
We were never refered to an allergician for eczema, we also took him abroad as somebody recommended this wonderful dermatologist (sure there are some in these country but nobody could point us in their direction) and all what I wrote (eucerin included) was what the prescription and comments of that dermatologist.
Now, when we came back with the diagnose from Spanish consultant we were taken seriously (finally) and refered to dermatology, DS even got a pass to bypass the GP and go straight to the dermatologist for a year which we never used, the treatment prescribed abroad has kept him free of eczema for almost 3 years now.
Your GP can order RAST tests (at your cost) to bypass the need of waiting for ever for an appointment with the allergician, but most avoid to do so as the knowledge of allergies for a regular GP is somewhat limited and they find it difficult to act on the results., or you can get her tested for intolerancies via an Elisa test that you can get privately but it's not particularly reliable.
Now, bear in mind that your baby is still developing and many of these sensitivities are very likely to disapear by the time she is one year old.