The place I would recommend is near Hatton Garden, which is the jewellery capital of London, go to Charterhouse Street. CSO Valuations, I'm afraid I don't know the phone number or the street number, but it should be easy to find. It is part of the DeBeers group, they specialise in cut diamonds and will probably be able to either value your stones or tell you the best place to go.
There are different types of valuation;
If it is for 'Probate' you need to have a low value.
If it is for insurance it is better to go for a higher value.
This is because of the different reasons for selling or buying; ie a forced sale can only fetch a price at the current time of sale, to a market that is willing to buy. Whereas, if you are valuing a diamond for insurance purposes you need to know the replacement 'retail' value.
If the diamonds are cut and not set CSO Valuations will be able to give you an idea of the current value, however, if they are set (such as in a ring or necklace etc), you will need to go to a jeweller for a replacement valuation.