Hi OP!
I suffered from severe hyperemesis from 4-25 weeks so I'm hoping I can provide some good advice, but please ask if you have any more questions.
Medicines during pregnancy are categorised in alphabetical order, with A being the 'most risky', B less so, C even less so etc etc
Cyclizine is around a C or D, which means the risk is very very very minimal but that for obvious reasons it hasn't been tested on pregnant women, but currently no known effects have been proven.
It will be the first choice for doctors to prescribe for the simple reason in that is it dirt cheap.
However, it comes with side affects such as increased sickness (annoying as that's what it is meant to stop) but all of these side affects effect you and not the baby, so please breathe easy in knowing that your baby is fine.
If you find it isn't working, there are multiple alternatives such as phenergen, metoclopramide and ondansetron.
Sometimes you might need a combination of every drug, as I did, in order to provide a proper relief from the sickness.
Some doctors, nurses and GP's will have very limited knowledge on pregnancy sickness, so you may find at times you really have to push for good care.