最終電車で寝過ごし、隣の駅から自宅まで歩いて帰ったって英語でなんて言うの？

This is probably the easiest way to phrase what happened to you. By using lots of commas to separate each part of the story, you can shorten your explanation to just one sentence.

I fell asleep and missed my stop on the last train. I had to walk back home from the next station.
最終電車で眠ってしまい、自分の駅で降り損ねました。隣の駅から歩いて家に帰らなければなりませんでした。

「寝過ごす」を調べるとoversleptという単語が出てきやすいですが、こちらは約束の時間より遅く起きた、朝寝坊や昼寝をしすぎた場合に使われます。自分の駅やバス停を降り損ねたのは missed my stopという表現の方が一般的です。そもそも電車で眠るという安全なことができるのは海外ではレアケースなのだと思います…

walk back home で「歩いて家に帰る」という表現になります。

If you 'take a nap' as stated in the first statement, it means that you sleep for a short period of time. 'Taking a nap' usually happens during the day either voluntarily or involuntarily because of tiredness. 'Taking a nap' involuntarily means that you 'unexpectedly fall asleep'. It means that you did not plan to sleep but it just happened. If you are on a train or a bus, you are bound to miss your stop unsless someone wakes you up which rarely happens. The result is that you miss your usual stop and have to 'alight' at the next station or stop. The verb to 'alight' used in the first statement means to 'disembark' or 'get off' a train, bus, or airplane.
So, you may say:
I took a nap on the train while going home and missed my usual stop, resulting in me having to alight at the next train station and walk home from there.
or
I unexpectedly fell asleep on the train while returning home from work, resulting in me having to get off at the next train station and walk home two kilometers away from there.

I fell asleep on the train and missed my stop so I had to walk home from the next station. - You "fall asleep" unexpectedly or involuntarily due to exhaustion. You can also say, "I dozed off" or "I nodded off" on the train.

I got off at the next station and walked home from there. - You can "get off" at a stop or "exit" or "alight" the train/bus or stop.

In order to sum up the situation, starting with a brief statement of the end result (walking further than usual) is a good way to lead to what the cause was. Dozing off is a good expression to use for indicating that one fell asleep unintentionally.

You were going home on the final train of the day. But you fell asleep and missed your stop. So you walked home from the next station (one station from your home). You wish to explain that. In that case, one of the above suggestions should be fine.

If you was on your way home on the final train of the day, and you fell asleep and missed your stop. This made you have to walk home from the next station, you could express this by saying either of these statements: "I fell asleep and missed my stop on the last train. I had to walk back home from the next station" or "When I was on the train I fell asleep, and I woke up at the next stop. I had to walk home from the next station".

Example:

When I was on the way home from work on the last train of the day, I accidentally fell asleep and missed my stop, so I had to get off at the next stop and walk home from there! That's why it took me such a long time to get home!

When explaining a slightly long story, you can shorten it down by using commas.

Get off - means exit the train, had to leave.

It was the last train of the day, but i fell asleep and unfortunately missed my stop. I had to get off at the next station and walk.

You could use this sentence above to explain the following situation.

The first two examples given are exactly the same in structure and words with the key difference being between the two verbs, "to fall asleep," and, "to doze off," which can be used interchangeably. The last example doesn't explain why you, "missed the stop," but still provides all the information as the first two.