In most Bulgarian mini markets/supermarkets/childrens' stores (ÃÂâ€ÂõтÑÂÂúø ÃÂœðóð÷øýø/Detski Magazini)/chemists (Aÿтõúø/Apteki), you will find the usual multinational/international brands, such as Kraft, Heinz, Pampers, Nestlé, etc., etc., as well as Bulgarian brands.
As Sunny Beach really only exists as a town in the summer and is mainly for tourists, I'd be surprised if you don't find at least some names that you recognise, if not 'English' brands.
Baby food in Bulgarian is 'Ã¥рðýð ÷ð ñõñõ' (Hrana za bebe), baby milk is 'üÃÂȄÂÂúþ ÷ð ñõñõ' (Mlyako za bebe).