The concept is good - the details need work.
You know that :-).
Maximum charge voltage:
If using NimH batteries, which are rated at a nominal 1.2V, charge voltage for full charge is in the 1.4V - 1.45V range at charge rates of up to around C/10 (mA = mAh/10).
Using two diodes as shown as a 'clamp' will reduce capacity stored to a small total of full capacity. Better is to use a well defined clamp such as a TL431 "adjustable zener" - plus an extra clamp transistor if power levels warrant. Details of this and other options can be provided once the other details are sorted out.
Here is a TL431 + PFET clamp regulator circuit - from my SE EE answer here.
R2 can be a wire link.
For currents below TL:431 rated Imax remove Q1 and short R2 & R1.
To use a small pnp bipolar transistor in place of Q1 use R1 = 10k, R2 = 1K.
For NiMH adjust Vclamp to about 1.4V/cell.
For LiIon adjuist Vclamp to 4.0V/cell or slightly LESS.
LEDs require a minimum forward voltage (Vf) to operate.
Red requires about 2V,
This table is 'somewhat generalised' but gives an idea.
Modern white LEDs typically need 3V or more. If you have control electronics then you also need some "headroom voltage". If you are operating white LEDs directly from battery then 3 x NimH will work acceptably and 4 is probably better.
An alternative is to use a single cell Lithium Ion (LiIon) battery - such as are used in almost all cell phones. These are readily available and even a well used one may meet your need. The key thing about LiIon is that they MUST be charged in such a way that they are not rapidly destroyed. This is easily enough achieved but must be done.
Alternatior / Generator:
A small brushed DC motor rated at 6V or more will probably meet the need.
Small stepper motors also can work well - but may have somewhat more startup torque.
Small brushless DC motors can work very well. Surplus low voltage DC fans form PCs etc are a good source - but the internal electronics need to be removed or bypassed. That's not hard and there is much on web re doing this. More on that if required.
The stepper motor and brushless motor solutions will usually have longer lifetimes as there is no brush wear.
If you want to use a single NimH cell then a DC-DC step up inverter will allow LEDs to be driven from a 1v - 1.2V source. These can be built 'from scratch' but salvaging one from a single cell lawn light is a low to no cost option. Lawn lights may die for a range of reason so a dead one may be the source of a suitable inverter.
Wind turbine mechanical aspects are not hard when power levels are this low and efficiency is not very important. Ask if needed.
See what you think of the above and ask questions and the answer can be expanded to suit.
An excellent charging solution if using a LiIon cell is to use a TP4056 LiIon charging module. These are available for a dollar or few on ebay, Ali Express etc. There are versions with and without a low voltage cutout. The ones with cutout are slightly costlier but protect the battery from over discharge. A white LED operated from a LiIon cell will usually reduce current drain into the microamps range while the battery is still in its 'safe' voltage range.
Here is an EXAMPLE ONLY TP4056 charger module with low voltage cutout (seperate output and battery terminals). from here