I've found they all carry different items and all have different stock levels.
I've had the best customer service at the department stores, so both Harrods and Selfridges - the SA's there really couldn't help enough. The stand alone stores, on the other hand, I found were awful, but had more stock.
New Bond Street staff really were not attentive, but they seem to have the best stock levels as it's the flag ship store.
Sloane Street I found to be a bit of a mix, not a very busy store, so usually has quite good stock levels. But I've found some of the staff to be a bit hit and miss.
Sorry if that isn't much help, I guess it's up to you whether you want the customer service! But personally, I would recommend the Selfridges and Harrods stores for you to definitely try and purchase from first, and if that fails, try Bond Street.