To try and clarify some of the points above: The OP, Subbu, has a low pressure stored water system. The water coming out the taps and showers depends entirely on gravity for its flow.
To improve the flow (and 15 litres in 10 minutes is awful - you'd ideally want 15 litres in ONE minute) the single best solution is therefore to raise the height of the cold storage tank. DO THAT IF YOU CAN!
The flow will always be worse coming out of showers compared to taps (faucets) because the showerhead will be usually be held at a much higher level, so closer to the height of the cold storage tank. It's the DIFFERENCE between the height of the storage tank water level and the water outlet level that determines the amount of flow. Ie - if you were to keep lifting up your showerhead until it was at the same height as the tank (up in the loft!), then the water flow would stop completely. If, on the other hand, you took that showerhead down flat against your bathroom floor, you would have a much improved flow - as well as a mess...
Ok, if it's not possible to raise the height of the storage tank (it's already as high as it can go), then you should look at whether the pipes are routed as well as they can be - do they make the most direct route to the bathroom? Do the pipes have a slight 'fall' along all its length, or are there any horizontal runs? Do the pipes go first down to a lower level (say under your bath) before then coming back up to the shower? And - yes - going down to 15mm pipe is NOT good as this is a 'low pressure' system, so pipe sizing has a greater effect. All these things add additional 'friction' to the water flow, and - since it's driven at a low pressure anyway - this friction does make a difference. (Mains-pressure systems don't suffer like this at all - you can run it all pretty much in 1/2" pipe and up, down, and all-around...)
Ok, I don't think I'd entertain going up to a 2" pipe (that's huge), but an increase to 3/4" or even 1" pipe should help (although it depends on other factors as to how MUCH help).
Question for Subbu (although he's probably long gone) what water 'head' do you have? Ie - what difference in height do you have between the showerhead and the storage tank water level?
And, as said above, if raising the tank height and/or improving the pipe size/route cannot be done, then a pump is the only answer (without changing to a completely new, mains pressure system).