Hi and welcome to the forum,
I don't know much about plumbing and I've never tried taking water from a river, and I'm sure someone will come along with better information, but my thoughts are as follows:
The pressure will be defined by the height of the inlet. The higher the inlet compared to the outlet, the greater the pressure, as the only force in the equation here is gravity. To up the water pressure, you'd have to install some kind of inline pump I would think. What sort of fall are we talking about? From that, you could calculate the maximum obtainable pressure.
You could install some kind of tank, like a water butt, at the allotment end to make sure that you have a good supply of water even when the river runs drier in the summer, or simply to make it easier to fill up a watering can. The hose outlet could have a valve in it to stop the water flowing when the butt is full.
When the water arrives at the allotment end, assuming you have decent pressure from the fall, you could downsize the pipe to get a higher water flow rate which might allow you to get some kind of jet going to aid watering.
At the inlet end, you probably just need some kind of gauze box to stop anything too big getting into the pipe and blocking it. Whatever you do you'll probably need to clear it out reasonably regularly to stop it getting clogged. If the area is prone to freezing, you should also make sure you empty the pipe in the winter (doubt you'll need too much water for the allotment then anyway!) otherwise you could end up cracking it, unless you intend to bury it deep.
I don't really know about the pipe size, probably about an inch?
Hope someone with some experience comes along soon with some better answers.