The phrase "up to our necks" conveys a sense of being overwhelmed or deeply involved in a situation. Some synonyms for this phrase include "in over our heads," "drowning," and "swamped." Another alternative would be "snowed under," which is often used in professional settings to represent being overwhelmed with work or tasks. Alternatively, the phrase "in the thick of it" conveys a sense of being deeply involved in a situation, particularly one that is chaotic or intense. Other possible options might include "deeply immersed," "entangled," or "embroiled." Regardless of the specific phrase used, the sense of being deeply involved and challenged remains consistent.