Depending on the type of replacement hose you have...
There is/was a supple corrugated hose available that could be fitted without having to remove the head or the water pump. They were a fiddle to fit but did go on in the end. Normally come packed on a bit of card with two band type clips.
If you have a more durable, OE quality product it may be wise to remove and refit the water pump. I know that works on the inline engine cars like Morris, MG AH etc. The Mini is of course a tight squeeze.
Why must you have a bypass hose? The type of thermostat that has a blanking plunger to close off the bypass is usually never fitted. So the bypass hose only serves as a vent/bleed hole to release trapped air from the water pump and to divert water to or from the block when the thermostat is closed.
Later model cars like the Metro, Maestro, Mini continuation model, dispensed with this bypass arrangement and blanked off
both pump and cyl head bypass connections with rubber caps. How do they get
round the bleeding of the water pump or is it of no consequence?
On the Triumph 4 cyl TR we have for years had great difficulty getting a thermostat with a bypass blanking system (It closes the bypass hose connection when the thermostat is open and allowing full water flow into the radiator) Without bypass water control the engines are prone to overheat. So we block the bypass hose, (which is almost the diameter of a Mini main radiator hose), completely with a cork or brass bullet that has a 3-5 mm bleed hole drilled through it to allow any trapped air to bleed out. This works a treat and stops cooling water taking a short cut from the water pump to the top hose without cooling the engine.