There is a general principle that one should not engage in eating in the time immediately preceding the performance of a Biblical Mitzvah, as one may become engrossed in the meal and forget to perform the Mitzvah. The generally accepted Halachic amount beyond which one should avoid eating is a Kabetza. Similarly, the Shulhan Aruch (Orah Haim 235:2) says that one shouldnot begin eating a half hour before the time of the Shema at night, that is, at Tzet Hokochavim. If one started eating within that half hour, one should recite the Shema at its proper time and one may may continue one’s meal. After the meal, the Shulhan Aruch continues, one may recite the actual prayer of Arvitand repeat the Shema with its blessing.
If one must eat within that half hour, ideally one should ask someone else to remind one to recite the Shema at its proper time. If this is not possible, many Poskim are of the opinion that setting an alarm on one’s phone, watch, etc. would also suffice.
The Ben Ish Hai (in the newly published Mekabtziel) discusses a challenge he had in Bahgdad in which the congregation prayed Friday night Arvit early. Since they did not recite Sefirat Haomer or Keriat Shema (that is, they recited Shemaduring the prayer but not the Shema at its proper time) because of the early prayer time, it was questionable whether they would be able to start their Shabbat meal. He suggests an original idea that in such a situation, one may start his meal provided he does not eat more than a Kabetza of bread, prior to reciting the Shema. When the time comes, one should recite the Shema and Sefirat Omer and then resume one‘s meal and eat more bread.
Summary: One should not eat more than a Kabetza of bread within a halfhour of reading Shema at night.