I'm on a budget so if you also have recommendations on pieces to look for when thrifting/vintage stores/factory outlets or sales to look out or, that would also be very much appreciated.
The first thing you need is a Barbour wax cotton jacket, I prefer the Beaufort. You want to make sure you buy one that is actually made in the UK not the overseas model. Additionally I would look for one with a zip in liner for colder weather. They run around 400 but can be found on ebay this time of year for 100-150. The Bedale is also a good option but I like the hand warmer pockets on the Beaufort. You can wear this jacket in all kinds of weather with anything, especially in England.
It rains more where you live than in London. And extremes of heat or cold are not common although you may feel chilly compared to Malaysia.