How fast will my credit score increase?

have roughly a 1.5k credit balance currently, with 3 lines of credit. 2 being credit cards, with a combined cap of 4k.

You are using 37.5% of your credit just in carry a balance. Paying it off now will save money and improve your score. When you carry a balance it means that everything you put on the card is instantly charged interest, there is no grace period. The advice is to keep your utilization below 30% but that includes both your balance you are carrying plus new charges. Most people with high scores keep it below 10%.

I have 12k in students loans (all of which isn't required to start making payments until Q4 2021)

If you have money to pay this off quickly, then do so. Especially if interest is accruing. But if the interest rate on the credit cards is higher, pay the credit cards off first.

If you still have the student loans then the mortgage lender will figure the amount you are required to pay each month when determining if you can afford the mortgage. The more you owe in credit cards, student loans, and car loans the more difficult to get the mortgage approved even if at first glance the combination of down payment, and income would appear to support the monthly payment for the mortgage.

My current plan of action is to wait until I have the capital to purchase a home, and pay off my loans all at once. My question is, is this the most optimal strategy? If not, what would be?

You haven't mentioned your down-payment. It might make more sense to divert some of the down payment savings to pay off the debts, even if it delays your plan a few months. It might even save you money now.