Amazon.co.uk has for some time offered same-day evening delivery in urban areas of Greater London (within the M25) and Birmingham--check details on their website--for a fairly-stiff minimum charge. Local Amazon.shops could share whatever infrastructure supports same-day delivery. FWIW, it seems that in the UK local courier service (using motorcycles) is comparatively cheaper than in most major US cities, and much more commmonly used even outside the legal and financial businesses, so this idea could be expanded.

Other possibly-relevant factors:

Logistics of delivery in the US are, on average, much simpler; both UPS and FedEx are superlative (and Amazon uses such private carriers exclusively in much of the US), and there are good contract-delivery options for heavier and bulkier objects. Delivery locations (both residences and offices) are mostly accessible, being on average much newer. It's common for customers to take delivery at workplaces, and use their cars to take objects home.

By contrast, in the UK all the logistics companies are marginal performers, partly because the government's subsidized Royal Mail continues to command some usage. Much of the housing stock is vastly older and less easily accessible than in the US; heavy and oversize deliveries are often very much a matter of engineering custom solutions. Homes and offices can be far apart, and the customers often commute between by train, so there's no convenient auto to carry things home that were delivered to the office.

All these considerations may have contributed to the model of ordering over the web and picking up from a physical location. This model has been very rare in the U.S., but is not so uncommon in the U.K.