Is it obligatory for someone who was required to perform ghusl but was unable to do so until a period of time had passed, and instead performed tayammum in place of ghusl, to make up [the missed prayers/acts of worship]?

If the worshiper has a legitimate excuse that prevents him from performing ghusl (ritual bath), then his prayer with tayammum (dry ablution) is valid, and he is not required to repeat it. However, if he does not have a legitimate excuse, such as shyness or mild cold, then he must repent to Allah and repeat the prayers he performed with tayammum.