Sorting that out should be the first priority, and would be MUCH cheaper than any other solution to boost the hot water flow rate.
Find out where the cold water storage tank is. You'd expect low pressure froma combination cylinder, with the cold tank (feeding the hot cylinder) located just below the ceiling, on the same floor as the sink.
You could expect reasonable pressure and flow from a cold water storage cistern one storey or more above the sink. Maybe they've used pipes that are too small (cheaper) or they've used an imported tap designed for an European mains pressure supply. Either of those would restrict the flow excessively. The EEC import tap problem is tediously common. It could be a blockage or a partially closed service valve.
I will certainly investigate the overflow, but dont think there is one. The kitchen overlooks a sort of very public paved area and the flat wall is faces the main shopping centre of a busy new town..could they have not put an overflow in to stop it pouring onto passing people and taxis etc? It might have been a planning or aesthetic issue? /we are not able to have a tumble dryer vent in that wall without a lot of expensive planning hoo ha with the leasehold managers
Sort that out, if it can be sorted, before progressing to clever taps.