I got to C&W with a Uber
Even though we didn't have a worried look on our faces (I was 35 weeks so we were not sure whether that was actual labour, only lost my waters... Yes I thought maybe it's just nothing..) Still took a cab in the middle of the nigh to the Hospital, me with my big pregnant belly, partner with overnight bag, so I would assume the taxi driver guessed what was happening and he didn't say anything. Or maybe just 'Good luck'. I don't think a taxi driver would refuse you, especially for this occasion ! As for waters breaking, take a towel with you so if it actually happens in the cab, you can act on it quickly. All the best x