A coat seems reasonable, Steve, as Llewellyn could grab that and put it on as he walked. Whether he would have needed - or taken the time - to layer up with vests, waistcoats, jackets etc., with a PC waiting to escort him to the murder scene, is another matter.
There was virtually no rain that night, hence PC Thain not needing to wear his cloak, but there had been a thunderstorm the previous afternoon. And generally it seems to have been very rainy around that time (but then, it is England), as this clip suggests;
Lloyds Weekly 2nd Sept
"FRESH FLOODS IN THE EAST-END.
As a consequence of the heavy rains of the past few days there has been a renewal of the floods in the East-end of London, and on Wednesday many of the inhabitants of the Isle of Dogs, Plaistow, and Stratford were again compelled to leave their houses. The West Ham fire brigade were called out at an early hour to pump out the water in the basements and cellars. The great tract of grass land lying east of Stratford was again submerged, the cattle and pigs finding nothing but a mere fringe of grass to feed upon. Wanstead-flats were in parts almost impassable. The crops in Essex have been seriously damaged by Tuesday's and Wednesday's rain, and much of the wheat which had been stacked will be unfit for use."
This period of heavy rain also seems to have been what led to the terrible hop harvest that year.
That was one of the reasons I gave figures for 1-5 minutes possible delay.
Made basic assumption that he would put on trousers and some top cover, which could just be a coat over nightwear I guess
Just no way of knowing. Of course
Yes rain here is always a real possability I doubt that has changed much.