The definition of time 0 depends on the goals of the study. That often, but not always, is the date of recruitment into the study. Use the goals of the study and your knowledge of the subject matter to determine how time 0 should be defined.
In terms of how to deal with a covariate "collected at various times after recruitment," recall that the Cox model assumes that the hazard at any time is a function of the covariate values at that time. Survival calculations are done only at event times among the cohort being analyzed; the covariate value for the case having the event is effectively compared against the values of all cases still at risk but not yet having the event.
So if a covariate is measured so long after recruitment that it can't reasonably be considered to represent the value at recruitment, it can't be used for survival analysis at times prior to its measurement. You will have to use your knowledge of the subject matter to determine whether the covariate being measured is likely to change that quickly. It might be possible to impute a value for earlier times, based on details of your study design. It certainly can be used for survival analysis at times after it has been measured. If there are no events in the cohort before the time that the covariate is measured, there won't be much difference how you handle it (except for the definition of the baseline survival curve), as its value at the first event time will be the value needed for the first survival calculation.
That structure of calculations for Cox models also means that if a covariate is being measured at several times on the same individual then it should be treated as a time-dependent covariate. That is, if the value of the covariate changes for an individual later in the study, its value has to be updated for all subsequent times that individual is at risk for the event. There are standard ways to structure and analyze data sets having time-dependent covariates. That approach also might help you to handle a situation in which the covariate is only measured once, but so long after recruitment that it doesn't represent the value at recruitment.
Added in response to comment:
In this case, all but one of the covariates is measured at recruitment into the study, but for technical reasons one covariate is measured at some single time after recruitment. For specificity let's call the late-measured covariate C and say that it is measured at some time between 90 and 180 days after recruitment.
There is no general answer to how to proceed, as the answer depends on the goal of your study and the relative timing of events and measurements of C. In the above example, if all measurements of C were made by 180 days and there were no events in the cohort before 180 days then there would be no problem (except for the implicit assumption that the value of C will be constant into the future, as the Cox model is based on instantaneous values of covariates at cohort event times).
A problem arises if some individuals have the event before their values of C are measured. That means the population on which C was measured is inherently biased: it was only measured on those who avoided the event for the 90 to 180 days needed to get the measurement of C. Those with values for C thus had lower risk of having the event than the population that originally was recruited. That's a form of survivorship bias.
In that case you will have to use your knowledge of the subject matter to choose how to proceed. I suppose that if the event is not fatal, the event itself is known not to affect the values of C, C is ultimately measured on all participants, and values of C don't change substantially over time, you could use late-measured values of C to represent the value at recruitment time.
Alternatively, if having C in your model is important yet there is a risk of survivorship bias, you will have to acknowledge that the model with C only represents a low-risk subset of the full population. With that acknowledgement you could perform survival analysis with time of measuring C as time 0 for a survival analysis, as your colleagues propose. Whether that reference time makes any sense for your study depends on the nature of your study.
Using the date of measuring C as the time = 0 reference wouldn't by itself pose a problem for a Cox multiple regression (what many call "multivariate"). If you base a Cox regression on covariate values measured at date of recruitment and don't re-measure them later, then you are implicitly assuming that those values haven't changed at any time at which events occurred in the cohort. Insofar as that assumption is true, then you can simply use the covariate values measured at recruitment time as the values at the time of measuring C. If that assumption isn't true, then the Cox modeling itself is suspect.
