Is it worth it to upgrade training rooms?

Just to put some numbers here:

Strength Level | Base    | 1st Upgrade | 2nd Upgrade
----------------------------------------------------
1 >>> 2        | 26m     | 25m         | 24m 
2 >>> 3        | 1h 20m  | 1h 16m      | 1h 13m
3 >>> 4        | 2h 40m  | 2h 32m      | 2h 26m
4 >>> 5        | 4h 27m  | 4h 14m      | 4h 3m
5 >>> 6        | 6h 41m  | 6h 22m      | 6h 5m
6 >>> 7        | 9h 21m  | 8h 55m      | 8h 31m
7 >>> 8        | 12h 28m | 11h 53m     | 11h 21m
8 >>> 9        | 16h 2m  | 15h 17m     | 14h 36m
9 >>> 10       | 20h 3m  | 19h 6m      | 18h 15m

Source: http://www.ign.com/wikis/fallout-shelter/Weight_Room. Other rooms have very similar (though not identical) numbers.

So you should be really only updating those if you have some ungodly amount of CAPS and it's burning a hole in your vault.

Update

I have some 700k caps now, and I really don't know what to spend them on. Perhaps, now might be the time to upgrade some training rooms. However, more than half of my dwellers have full SPECIAL now and training rooms are half-filled, at best. Again, upgrading training rooms wouldn't have any effect, other than making me 50-100k caps lighter.

Regardless of caps costs, you shouldn't upgrade training rooms. The reason is that higher level rooms get more severe incidents. The dwellers in your training rooms are usually low level, so if you don't upgrade your training rooms your rookies will be able to handle incidents unattended. If your training rooms are upgraded, you will have to heal your dwellers during the incident to keep them from dying. So even once you get more caps than you need, you should avoid upgrading your training rooms.

Later on, with large vaults, I'd say time is more important than money, and every little bit helps and are well worth upgrading the training rooms. Especially once you realize that having a Dweller's Endurance maxed out BEFORE they start leveling up drastically increases their hit points and start trying to max out level 1 dwellers before sending them to any tasks. At that point, the wait can get painful.

That being said, from what I've been able to find, it seems like the # of dwellers in a room training at the time may have a larger effect than the level of the room.

I haven't been able to find any greatly detailed or thoroughly researched numbers, but based on this reddit post and accompanying spreadsheet going from stat 9 to 10 in a tier1 room vs tier3 room is only a savings of 1 hour 18 minutes. Whereas having a tier 2 room with 6 dwellers in it vs just 1 dweller going from 9 to 10 is a savings of just shy of 2 hours.

So, it may be more beneficial to build 3 block training rooms and try to have 6 dwellers training at once, vs paying for all the upgrades. Of course, having 3 wide, fully upgraded training rooms with 6 dwellers training is going to give you the shortest training times possible.

Build across then in. It's much cheaper to build across 2-3 rooms than it is to upgrade. You DO want to upgrade though, it increases your storage which is your main buffer against disaster. Also by building across first you save yourself the grief of having to delete a room just to expand one you really need.

Up until this point in my progress. All Ive done is upgraded my vault little by little, collecting a few caps, couple of uniforms, and some pretty decent weaponary. Currently on a new game. I've got 25 dwellers, I'm able to maintain my power, food, and my water all at full capacity. I've already got Lucas Simms legendary dweller. He came to me right at the beginning of the game at level 36 in a lunch box. He's level 50 now. My max level for my common dwellers as of now are somewhere around 18-24. I've also taking the time and put in quite a bit of effort to upgrade my Gym 3 times, and let me just say WOW! that took for ever. I sent my legendary dweller out into the wasteland roughly for about 2 days 6 hours. I simply ran out of stim packs. I didn't want to risk funding such an attrosish fee of caps for Lucas Simms an his stats, all 80 of the items he found out in the waste land. So I brought him back before he made it to the armoury out in the waste at the 60:00 hour mark. Well, at least all 25 of my dwellers are now fully locked and loaded, and fully equipped with the nessacary uniforms in association to their corresponded rooms. Plus many extras. I've been taking this slow as in any fallout game. You simply just can't take it on all at once. Don't be so quick to upgrade your power, water,and food rooms with. First build 3 rooms side by side, then fill them up with dwellers. My suggestion on the second step is then to upgrade every room. Don't buy the next room simply cause you have enough dwellers to build it. Just know that it will cost power which is a total nuisance to try and cope with. Especially if you don't have enough resources to do so. When this happens we sometimes are forced to rush and what tends to happen quite often is it just failed. And let me state thatfew rushes which halts production and puts everything into chaos, not only slows the timer down but damages your dwellers, especially if your rushing in a fully upgraded room. Not having stim packs will put them into jeopardy because say you rushed. And a fire breaks out, they get hurt, u rush again that fails too, and you do it one more time to see if you can't get it to just magically work for you and nope, that failed too. Now if a fire or infestation happens in that room even though you didn't rush it, and you don't have the stims to revive them, this especially near that start of the game is gonna kill you. I even almost gave up a few times myself but trial and error. So Before you rush, be prepared, always provide the adiquit resources for every scenario in game. Health also has an impact and also regulates to some degree of a dwellers happiness. All I know is for me and my 25 dwellers, happiness is finally raising to 100, four of my dwellers have a strength of 10 only after a few days of waiting. Trust me it will all be worth it. These are going to be some fierce breeding grounds in the next couple days. Lol. I realize I need dwellers, in order to have dwellers I need food and water. Well time to impregnat around 10 females and get my next two rooms in order. A new fully upgraded diner and water treatment plant definatley will require an new power plant. So that's going to be 18 people I've got to be ready to provide for before I even build that stuff. The training rooms lol. I've grasped the concept of this game and I find this really enjoyable. As long as I'm able to continually have in my possession stim packs I should be fine. Rad always are God to have to, I will figure out later just how important they really are. Please Forgive me if this isn't in depth all that much I realize this, I'm just posting my findings so far on this obviously fun and addicting game. I just started playing this after all.