The wearing of Motocycle Helmets is covered in the UK by the Motor Cycles (Protective Helmets) Regulations 1988 no 1807. specifically section 4 - Protective head gear
Due to the wording of this act a helmet must be worn on a motorcycle or as a rider of a motorcycle with a side car, However a motorcycle in this regulation is classified in subsection 3 as 'motor bicycle" means a two-wheeled motor cycle, whether or not having a side-car attached'
So in other words you do not have to wear a helmet whilst riding a Trike.
However as many people have indicated that should you be unfortunate enough to have an accident on your trike, some insurance companies may try to use against you, e.g if you sustain head injuries they may argue that you had failed to protect yourself therefore you took responsibilty.
As far as the Police re concerned, do no more than to print the regulations out from here, and carry it with you. If you get pulled and quizzed about your lack of lid, just preasent them with your copy . Do not alter it in any way, as it is an official government version of the document and will be recognised as such.
Protective headgear
4. - (1) Save as provided in paragraph (2), every person driving or riding (otherwise than in a side-car) on a motor bicycle when on a road shall wear protective headgear.
(2) Nothing in paragraph (1) shall apply to any person driving or riding on a motor bicycle if -
(a) it is a mowing machine; or
(b) it is for the time being propelled by a person on foot.
(3) In this regulation -
"motor bicycle" means a two-wheeled motor cycle, whether or not having a side-car attached, and for the purposes of this definition where the distance measured between the centre of the area of contact with the road surface of any two wheels of a motor cycle is less than 460 millimetres, those wheels shall be counted as one wheel;
"protective headgear" means a helmet which -
(a) either -
(i) bears a marking applied by its manufacturer indicating compliance with the specifications contained in one of the British Standards (whether or not as modified by any amendment) mentioned in Schedule 2 to these Regulations; or
(ii) is of a type manufactured for use by persons on motor cycles which by virtue of its shape, material and construction could reasonably be expected to afford to the wearer a degree of protection from accidental injury similar to or greater than that provided by a helmet of a type prescribed by regulation 5;
(b) if worn with a chin cup attached to or held in position by a strap, is provided with an additional strap (to be fastened under the wearer's jaw) for securing the helmet to the head; and
(c) is securely fastened to the head by means of straps provided for that purpose; and
"strap" includes any fastening device.
In the Republic you only have to wear a helmet when driving a "Motorised Bicycle" (Motor Bike). You do not have to wear a helmet on a Trike as it is not a "Motorised Bike",but a "Motorised Tricycle". Having said this remember what a stone does to a windscreen at 60 miles per hour. The same could happen to your head. No point looking cool laid out on a slab.