Hi there. 

I'm may be running into an issue in my homebrew campaign I'm am looking into the possibility of tweaking the proficiency training. As far as I know i, according to Xanathars, it take 10 work weeks (5 days) - intelligence to learn a new skill/ tool etc. 

Now while I realize that his has been greatly trimmed down from the DMG, I fear that even this is too long, especially when we may want to move on and we just don't have the time.  I was thinking of doing something like this. 

You need someone who is an expert in the field your looking for, be willing to pay for their time (money or a quest) and lastly roll a d12 - intelligence. Each number of the dice is a day. If it is a 1 they re-role the dice. 

After which they become trained in that tool, or have gained +1 to history or now know how to weild that greatsword etc.  Gaining a proficiency will require to roll yet again. Is this a reasonable thing to do? What are your thoughts? 

If you’re the DM, you can just do it how you like. I think the  rules are there basically to give a DM a back door to control players doing it, since they can just not allow enough downtime. If you are ok with players doing it, you can just hand wave the time they spend doing it. 
I think really almost any amount of time can work, depending on the needs of the campaign, just be consistent. And consider that if you make it too short, players will do it a lot. 

Also, do you mean literally +1 to history. Generally proficient in a skill means add you proficiency bonus, which will be at least 2. 

Sounds like it would be a +1 instead of proficiency bonus.  At that point they would roll again and get the full proficiency bonus. It does take quite a while to learn things, but it is really hard to get training other than from feats.  The feat method of getting skills/tools/weapon/armor proficiency is technically instantaneous so would be faster than this method.  Only game play balance issue would be when the DM does not control access to the experts.

With this you have a max of 4.8 (24/5) weeks and a minimum of I assume 2 days - could go higher with lower than 10 INT.  Though you have the player re-roll on a 1, but what happens when the Wizard with the 18 INT rolls a 2?

Oh, I see now. 
I guess the short version is just to give it to them if you like, and not worry about the details. 

Yeah I just went on a rant like yesterday about how downtime restrictions are a bad idea because in some cases it just costs 5 seconds and a wave of the hand and in other cases it's just not possible at all.

Time is not a meaningful cost unless your campaign has specific consequences for time passing. If you want a new skill to cost resources, I'd probably recommend some sort of treasure cost, or some kind of side quest where the character has to contribute a portion of their daily resources (like a spell slot or class ability) to something that can grant proficiency (a deity or patron or mentor) that could be done during the ongoing party adventures. So you have something that takes time, but has meaningful cost (you can't fight at 100%) and allows the party and story to keep going while you're learning.

You can also have periodic stretches of downtime in your campaign where the characters have several choices on how to spend their time. One of these could be training towards a proficiency, but there should be other enticing options for the downtime so that there's a meaningful choice there.

Most DMs don't make you train or pay gold to gain the abilites you would get when you make a level but, some might if everyone feels it adds to the game as far as realism is involved. It could be the same with learning additional proficiencies. RP it if it suits you or just set a level based requirement that all players can take advantage of. For example, every 6 levels earned, a proficiency can be added. That would provide a proficiency at level 7, 13,  and 19(or at 20 instead).

My thought is don't add crunch you don't have to.  Your players have as much downtime as you, the DM, want them to have.  10 minus INT modifier work weeks is fine to become proficient in using new tools and probably not nearly long enough to become actually proficient in a new language.

The real problem with downtime restrictions is that the DM either waves it or totally screws you over by being a dick head.

There are ton of times when if you say "No, I do NOT want to go out on another mission, I just got back in town and have things to do.  I am not a traveling murderer, I am a wizard that STUDIES. "  then the DM will kick you out, saying "the rest of the people want to go now."   It's five minutes of DM time and there is NO reason at all to take it away from the player.

Depends on the group. My current group will take hours on minor downtime activities. Considering we are set to play once every 2 weeks, that amount of time would be undesirable. Nothing wrong with a good hand wave once in a while.

Technically, skills cannot be trained, only acquired through class and feats. Only languages and tool proficiencies can be trained.

In most campaigns I am in, we can just time skip any stuff like this. If time does matter though, I would probably just shorten to 10 days, reduced by 1 day per int mod.