Mainly I use GUMSHOE rules, modified as needed for the setting. Here's a few things I changed that I can think of off the top of my head:
I cut the size of player character point pools down significantly (building characters from far fewer points in general than any iteration of GUMSHOE), but then made all pools refresh automatically at the start of every session. That way there's no need to remember from week to week how many points you spent in previous sessions. I figure CoNTINUUM has so much book-keeping via spanbooks and just general time-travel-related notekeeping, that I didn't want to have to waste extra effort and energy recording how many points are left over from last week. The smaller pools are to balance that.
Off the top of my head I don't remember exactly how many points I trimmed them down to, but I've frequently thought I could have trimmed them down a lot smaller than I did without any real consequences. Gumshoe characters get so many points, since the refreshes are only at the start of entire new mysteries/adventures, and with weekly free refreshes they only need a handful of points to advance the plot. If you trim it down too far, it's pretty easy to give the PCs bonus points later, or just let them train skills via normal Continuum training over time rules. Taking points away mid-campaign because you were too generous at the start would be rather harder.
In general, I found/ruled/converted that each Continuum skill rank above Novice is roughly equivalent one point of Gumshoe skill rating. I use 'clocks' more or less like they appear on the Continuum character sheet - each spend or roll of a skill gets you 1 step closer to the next level. If a character has no points to spend but engages in a roll anyway, they also get a clock in the skill, so it's possible to advance in a skill that you don't have or have spent all your points in already. It's not perfect, but it works well enough.
I had to add a couple of skills to cover things specific to Continuum, such as the various Aquarian Skills, and a few "better at spanning" skills to compensate for the lack of a Quick stat.
I made Gemini into a skill you could spend to have your Elder bail you out, which allows me to throw harder challenges at the player and still give them an "escape switch" for if I overdo it. These Gemini points go away when spent, and only refresh if your Span Rating increases. I did something similar with "Mentor" and "Fraternal Favors" so each PC started the campaign with a couple of "my Mentor shows up to save me" and a handful of "my chrony in the (insert specific fraternity here) owes me a favor that I can cash in right now' points. This gives the PCs a bit more narrative control than they would otherwise have in Continuum, but that's balanced by being Graceless (thus unable to spend any of those points because of too much frag, etc) feel more dangerous.
I also added a "Range" skill, of which all players have 50 points. Spending 1 point of Range lets you span a certain amount of time and/or distance, depending on your Span Rating. This was done partly to match other Gumshoe mechanics, and partly to reduce the amount of time spent calculating the exact minutes and seconds of shorter spans - while knowing your exact Age to the second is important from an in-character performance, we found it was getting in the way of keeping the plot rolling, and Range points helped make that faster. Full details at: http://transitivegaming.blogspot.com/2010/10/extreme-range.html
Besides adding things like I mentioned above, I simplified the gumshoe skill lists a bit. Most versions of Gumshoe have more crime scene forensics and evidence collecting type skill than Continuum really needs, but you may have to "season to taste" on that. There's a really light Gumshoe book called "FEAR ITSELF" that I recommend for this. Intended for less "CSI" style Gumshoe games, it has a smaller investigative skill list. It also has some good psionics-type skills that can be easily converted to represent your Aquarian Skills by just cutting out the creepy cthulhu bits.
I use Stability per Esoterrorists / default Gumshoe, and not any of the extra bells and whistles (sanity, sources and pillars, etc) from Trail of Cthulhu. I also added Continuum-specific Stability challenges/penalties, an early version of which can be seen at: http://transitivegaming.blogspot.com/2010/10/stability-chart-for-gumshoe-continuum.html
Time Combat is basically the same, but with various Gumshoe spends and rolls instead of Continuum's reliance on the Quick stat. Incidentally, that seems to have made time combat a lot more balanced, whereas before it could be really hard on characters that were created with lower-than-average Quick.
Physical Combat is much easier because Gumshoe is not a combat-intensive system. I was a little worried about that at first, but honestly it's turned out just fine. Continuum's weird hit location rules were way crunchier than the original game needed, anyway. One thing to be wary of is the Gumshoe rules for wrestling over a gun or getting the drop on someone. It's just too easy to surprise someone in Continuum (span in behind them), so I had to eliminate that or else every fight would be a single roll.
Obviously, that's not everything I converted to make this game run, but it's a good place to start, and about all I've got time for today. If you've got questions, drop me an email or comment.