Unless you need vault copy birth certificates from Home Affairs. This is what takes the longest. To get a certified copy of a birth certificate from the UK GRO takes only a week or two.A quicker way might be to have a look at the UK Ancestral visa. If her dad was born in England her Grandfather/Grandmother might be too. this is a quick and relatively painless route to get in the UK, and there are no financial requirements. Just Visa costs and NHS Surcharges.
Applying for citizenship in Austria might be a long process, i know it is for alot of countries. The ancestral takes a matter of weeks rather than months or years.