Help identifying 50s dress watch

[HawkinsT]

I've recently inherited a Rolex and really know nothing about it, would anyone here be able to help with some details (e.g. what model it is)?

I've tried looking online, and it seems to be very similar to a number of cellinis, only I understand they were made from the 70s on.

It came in a Bucherer case, and the strap has 750 CB hallmarked on it, indicating it's an 18K Bucherer strap. The watch itself has no identification numbers between the lugs (I haven't had the back off for fear of damaging it).

On the back of the case is engraved the serial 1337344, which I understand means it was made in 1959 (I had been told my grandfather purchased it in the 50s). The only other writing I can see is 'Rolex', 'Geneve', and 'Swiss' written on the face. The face is around 31mm wide.

Anything anyone can tell me about it would be much appreciated (or if any more pictures or information is wanted please ask).