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There are some sources that would date 1 Timothy earlier (such as the 50s) and 2nd Timothy later in the life of Paul. One of the difficulties in discussing the dating of 1 and 2 Timothy today is the prevalence of modernist and liberal scholars in the 1800s and 1900s who deny the authenticity of 1 and 2 Timothy as genuine works from Paul.

Since we are believers and are able to state in the affirmative that Paul wrote both of these letters, we can begin to narrow down the particular date.

The letters could not have been written after Paul's death (although there is some possibility that they were written prior to his death, and then delivered after he died). Therefore we are looking prior to 68 AD.

The letters would not have been written while Timothy and Paul were still together in Ministry. There would have been no need for correspondence if they were united.

This gives us a couple of windows of plausibility for the dates of composition:

1. Paul writes to Timothy two letters during the same time when they were apart.
2. Paul writes to Timothy one letter, and then after being together again, writes a second letter.

We know that Paul met Timothy during a trip to Derbe and Lystra (Acts 16:1). Earliest this trip could be was mid 40s (Most date the Jerusalem council in Acts 15 as between 15-20 years of the resurrection), far more likely for this trip to be early 50s. If you subscribe to option #2, then an early window when Paul could have written 1st Timothy when they were apart during the period of Acts 17. Another window for option #2 is found during the time of Acts 19.

For those who are proponents of option #1, there are some tricky aspects to the dating that have to do with Paul's perspective on death. We know that Paul often was faced with the threat of death after his public ministry began. We also know Paul was in prison on multiple occasions when he was apart from Timothy. While it's possible that the reference of Paul to his own impending death may be related to his final imprisonment (mid to late 60s), it's almost completely reasonable to say that Paul wrote with death in mind during one of the many times prior to his final imprisonment. We know from his conversation with Agabus (Acts 21) that Paul was willing to die on his way to Jerusalem, even though his death wouldn't come for several years down the road. That's not conclusive evidence for a specific date of 1 or 2 Timothy, but it gives us pause to consider that just because Paul spoke of death coming, it automatically necessitates the final imprisonment of Paul as the confirmed undeniable dating.

This all opens the door for multiple dates that could be origin points for 1 or 2 Timothy (or both).

I'll close with a few quotes for your consideration from some noted scholarly biblical works - DA Carson and Moo in their NT overview state 55 as a date (for 1st Timothy) according to some:

"Another suggestion is that we should take the reference to Paul’s departure
for Macedonia (1:3) to be that mentioned in Acts 20:1, after the riot in Ephesus.
Timothy was with Paul again in Acts 20:4, but evidently Acts 20:2 covers quite
an interval of time, and there could have been a letter between Acts 20:1 and
20:4. J. A. T. Robinson thinks 1 Timothy may contain the gist of the charge Paul
gave when he gathered the disciples and exhorted them (Acts 20:1). He dates
the letter in the autumn of A.D. 55, when Timothy was quite a young man (cf.
1 Cor. 16:10–11, which Robinson thinks was written in the same year) and in
need of the kind of directions Paul gives in this letter.53 Not many have been
convinced by this argument (the date seems to most students to be far too early),
but it must remain a possibility."

Matthew Henry on 1st Timothy:

"The design of the epistle appears to be, that Timothy having been left at Ephesus, St. Paul wrote to instruct him in the choice of proper officers in the church, as well as in the exercise of a regular ministry."

Matthew Henry on 2nd Timothy:

"The first design of this epistle seems to have been, to apprize Timothy of what had occurred during the imprisonment of the apostle, and to request him to come to Rome. But being uncertain whether he should be suffered to live to see him, Paul gives a variety of advices and encouragements, for the faithful discharge of his ministerial duties."

This reveals that Henry was in camp #2 believing Paul wrote 2 letters at two different intervals and visits and time spent with Timothy in between. I think this option is the more viable one for an earlier dating of 1st Timothy, and thereby explains some of the distinction between the content in the letter to Timothy regarding what was going on in Ephesus, and why there was such a shift and drift by the time of Jesus's Revelation to John in Revelation 2. 1 Timothy on this timeline would be some time prior to the trials and imprisonment of Paul post Acts 20, making for at least 5+ years between 1 Timothy and Revelation 2, and at most as much as 10+ years.