I have a Dr.rer.nat from Germany but currently I am working for an institute in the United Kingdom. Is it legal or acceptable to continue to use the title Dr.rer.nat (First Name) (Last name) in the UK or do I need to change it to (First Name) (Last name), (Ph.D)?
In the UK the holder of a PhD (or other professional doctorate) will generally use the title Dr as opposed to Mr/Mrs/Miss that would normally be used.
Unlike in Germany there is no addition to field of study.
Therefore the correct title would be Dr (First Name)(Last Name).
It is certainly legal to refer to yourself by the title you would use in Germany - few titles are actually legally controlled in the UK - and I would say it is also acceptable. However, I would suggest that is advisable to use the normal UK convention so that your audience understands what your title means without you needing to explain.
I would therefore use Dr user23530 rather than Dr.rer.nat user23530.