Hi there,
most of the best shops have now been listed, another note that Tesco's international shelves will also have the basic items though depends on the store, Budaors Tesco seems to be the best. Tesco here also started online ordering and home delivery and it's in English too.
culinaris.hu in 5th and the Asian store next to the Grand Market on Vamhaz korut https://www.facebook.com/pages/Ázsia-Bt/539021799452655 are the best stores to find anything more exotic if you need stuff for a meal (spices other ingredients)
The main point is that most of these items will be a LOT more expensive than in the UK even if you can find them here so when we go we always bring back basic supplies of the below. Also not sure if you know but Hungary has a VAT of 27% the highest in the EU so that explains why things are so expensive here, also in the UK there is no VAT on children's clothes, baby supplies and shoes - so unless the sales are on here we shop for most of our clothes in the UK...
so we tend to bring:
- breakfast cereal, mueslis - in normal supermarkets you can't get normal cheerios for example only the very sweet honey ones, good muesli is expensive, good brands like my favourite Dorset muesli you pay a fortune
- set honey - Hungary has great honey but this is my favourite and we always bring some back
- basic spices for an Indian, Thai, mango chutney
- good cheddar cheese - selection here even in Tesco is pretty poor so we always bring back cheese
- if you eat pork - we do stock up on those good English sausages too for a good sausage and mash esp for the winter months
- if you arrive before Xmas and if you celebrate it - make sure you stock up in M&S for the Xmas puddings, minced pies and your xmas crackers, cranberry sauce etc - they would be hard to find and/or for double the money - one of the years the Great Market Hall organised 3 days of British market and we paid ridiculous money for some crackers we learnt our lesson since. Hungarian traditional Xmas won't have any of these
- my English husband always gets annoyed how you can't get easy over the counter medicine here - you can't get them in the supermarkets here only in the chemist and even then you have to ask for them and interact in Hungarian - we always stock up on basic supplies of paracetamol, ibuprofen, lemsip etc, children's medicine is also more strict here and very often you need a prescription from the doctor when the same thing you would just get from the shelves from Boots (a recent example when I tried to get some allergy relief for my 3.5 yrs old and was told I need prescription)
Will let you know if I can think of more :)
Good luck!
Andi