What is entailed by not completing the Ramadan fast in past years due to family pressures, and is it obligatory to make up the days despite not remembering their number?

You must first repent to Allah the Almighty, then make up the days you missed from previous Ramadans after reaching puberty. If you do not know their number, then strive to determine them until you are certain or have a strong presumption that your obligation has been fulfilled, because making up those days is a debt upon you, and "Allah's debt is more worthy of being paid." The Ummah has unanimously agreed that whoever intentionally misses fasting Ramadan (definition) and then repents, must make up those fasts.