Yes, normally if you are not at fault in an accident your insurance company will subrogate with the at-fault party or their insurance company. Since the person that caused the accident was driving without insurance than your insurance provider will likely go after the at-fault person personally for the monies they pay out on your claim. You can also speak to your insurance company about trying to recoup your deductible amount from the at-fault party.
After paying a claim caused by the negligence of another driver your insurance company will usually assign an adjuster to pursue the negligent party (or their insurer if they were properly insured) for reimbursement. Some insurance companies call this type of adjuster a Payment Recovery Adjuster whose job is to subrogate with the at-fault party and to keep the customer informed on the status of our payment recovery efforts. Other companies have a subrogation department (often called "subro") who takes care of this once your insurance company's portion of the claim is closed.
[Let CarInsurance.com help you find affordable auto insurance now.] On a typical claim, a the payment recovery adjuster will notify the at-fault party that the victim's insurance company is pursuing reimbursement of the amount they paid out on the claim and perhaps also your deductible amount. Next they will send claim documentation and proof of payment to the negligent party and basically a demand letter.
Usually the adjuster will have to negotiate with the negligent party until reimbursement is received or until all efforts are exhausted in trying to do so. If the insurance company is successful in their subrogation efforts for the entire amount that they paid on the claim and your deductible amount they should then send you a check for repayment of your deductible amount.
In some situations your insurer will not recover the full amount they request. This sometimes occurs when the other party involved disputes the cause of the accident or does not have insurance, like in your situation. While insurance companies typically make every effort to recover the appropriate amount they cannot guarantee recovery of your full deductible normally since they may not receive all of their monies they paid out either from the at-fault party.
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As you can tell the subrogation process takes time so payment recovery usually takes about six months. But with the other person being uninsured it would likely take longer. In some cases your insurer may not be able to recover anything at all.
Keep in mind that states have different rules regarding payment recovery, and some states do not allow payment recovery for some coverages. To find out what your state laws allow contact your state's insurance regulatory body. You can also find out if the state will suspend the driver's license of the uninsured motorist if they fail to pay your insurance company for the damages they caused when requested.
Speak to your agent or claims adjuster already on assigned to the claim to find out how exactly your specific insurance company deals with subrogation and if they will also try to recover your deductible amount. | 
