Brief Answer:
Read below.

Detailed Answer:
I read your question carefully and I understand your concern.

I must say first that there is no direct relationship between the lower back pain and tingling in the hands. The way the nervous system is organized from upwards to downwards back issues manifest symptoms from that level downwards, not in the upper limb. When tingling is related to the spine it is to the cervical spine not the lumbar.

So addressing them separately. Tingling in the hand can be due to nerve damage. It can be due to compression of a single peripheral nerve, the most common ones being the median nerve, causing carpal tunnel syndrome, with abnormal sensations such as tingling in the palm and first three finger. Other nerves can be involved as well.
There may also be nerve compression higher up like in the neck due to disc herniation or other degenerative changes of the spine, usually there is some neck pain as well though.
It might have been easier if you had provided some more detail regarding the area involved by the tingling sensation.
Tingling in the hand may be cause also by electrolyte alterations, thyroid dysfunction or anxiety, but usually symptoms are on both sides.
So if that tingling persists a physical examination for compression signs as well as tests such as nerve conduction studies may be necessary.

As for the lower back pain, the most common causes are muscle strain and arthritis. Initial management consists in over the counter anti-inflammatory medication like Ibuprofen, a muscle relaxant such as Flexeril for 1-2 weeks as well physical therapy later on, in order to strengthen spine core muscles and prevent further recurrences.

I remain at your disposal for other questions.

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Brief Answer:
Nerve conduction studies necessary.

Detailed Answer:
Thank you for the additional information.

The old injury shouldn't have anything to do with the current tingling. If symptoms isolated to the fingers not more upwards, no neck pain, then a peripheral nerve compression is more likely, carpal tunnel syndrome would be the primary suspect, compression of the median nerve as it traverses the wrist. As I said before the diagnosis should be confirmed through nerve conduction studies.
If it is indeed confirmed, your habits should be reviewed, wrist actions which might promote that compression. Some blood tests for causes such as diabetes or rheumatological conditions which are commonly associated with such peripheral neuropathies.

Let me know if I can further assist you.

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