Presumably you are well under age 70½. If you had sufficient compensation in 2018 to support a $5,500 contribution for 2018, yes, you can make a $5,500 nondeductible traditional IRA contribution for 2018 by April 15, 2019. It will be reportable on your 2018 Form 8606. The result will be basis in nondeductible contributions appearing on line 14 that will transfer to line 2 of your 2019 Form 8606.
If you'll have sufficient compensation in 2019 to support a $5,500 contribution for 2019, yes, you can make a $5,500 nondeductible traditional IRA contribution for 2019 by April 15, 2020. Note that the regular contribution limit for 2019 is $6,000, so you could actually contribute $6,000 for 2019 (given sufficient compensation). The nondeductible contribution for 2019 will be reportable on your 2019 Form 8606.
Assuming that you have money in a traditional IRA, you can do a Roth conversion whenever you wish. The Roth conversion is reportable on the tax return for the year in which the distribution from the traditional IRA occurs. Converting immediately after you make the 2018 and 2019 traditional IRA contributions means that it will be reportable on your 2019 tax return that you prepare a year from now.
Had you made the contribution for 2018 before the end of 2018 and converted it to Roth before the end of 2018, the conversion would appear on your 2018 tax return instead of on your 2019 tax return. Since converting immediately result in zero taxable income when you have no other money in traditional IRAs, there would be no difference in the taxable result had you contributed and converted $5,500 during 2018.
