I don't think getting paid in January changes anything (after all most people get their pay for December in the first few days of January). If you started working in December, you had income in 2019 so you are theoretically supposed to file a tax return for 2019. It would have been reported as such by your employer to the tax office. And if your employer withheld some money from your December salary, you might even be entitled to a refund.
However, if you failed to file a return in time, fear not, in France, you cannot face any consequence (fine, etc.) if you did not owe taxes. Since the first 10 064 € of income are tax-free (for a single person, more if you are married and your spouse has no income), unless you have a very high salary indeed (or some other source of income), you shouldn't be in trouble. This happened to me last year and I didn't have to pay any penalty.
The French tax office is also pretty lenient if you make a good faith effort at rectifying errors. So I would still try to file a tax return. If you cannot do it online as a new resident, you will have to do it on paper. The form is daunting but if your situation is simple (e.g. no other income than your salary), there is only a few boxes to fill in.
Finally, note that having a tax statement for 2019 can actually be useful: when you have to document your income in France, it's customary to present a tax return for the year before last. And if yours state that you didn't owe nor pay taxes, it could work to your advantage (more subsidies from your CSE, etc.)