As others pointed out, there are no difference in storage facilities across aircraft types - the storage conditions have to meet very demanding regulations and there is no way around them.

Apart from differences in suppliers, the only difference related to storage pertains to overnight flights that are catered in London (ie essentially short haul). Those have a simplified (hear - less good) breakfast due to the food having been stored overnight and will also exclude a small number of SPML options (such as KSML) as a result. Also on some short haul aircrafts, galley/storage space may have an impact on the amount of meals/size of trays that can be carried - so for instance, in theory, lack of space could bring some airlines to only take half trays instead of full trays (or quarter trays instead of half trays) due to lack of space on some small regional aircrafts, but again, that would be short haul only and to be honest, full trays become a rarity on regional flights anyway, plus the very demise of food in short haul Y on many airlines means more space for the few rows of seats that actually get a proper meal. In any case, I don't believe this issue affects BA Cityflier though I might be wrong.

Finally, different types of aircrafts do have different ovens and that can in fact have an impact on how good the food tastes! However, this does not affect what type of food can be served/stored/heated just its gustative quality and likelihood to dry out.