The actual size of the 700C rim/tyre is 622 mm diameter where the tyre locates against the rim. That's 8 mm smaller diameter than a 27 inch wheel/tyre which is 630 mm.
[Historically 700C was also known in northern Europe as 28 inch, which is confusing because it's actually smaller than British 27 inch (and there's a British 28 inch size which is 635 mm, still used on roadsters in Asia). More recently 622 mm was adopted for mountain bikes, but as they use wider tyres these became known as 29 inch.]

If you're on 27x1-1/4" then your effective tyre diameter is 630 + 2 × 25.4 × 1.25 = 693.5 mm. Subtract 622, divide by 2 and you get 35.75 mm, so switching to 700x35C will give you almost exactly the same rolling diameter as your 27x1-1/4" tyre. As the rim is 4 mm smaller radius your brakes will need to reach 4 mm further, which is within the range of adjustment for some brakes, otherwise you'll need a caliper with a deeper drop.
[Note that 27x1.375" looks like it is the same numbers as 27x1-3/8" but the decimal version is incorrect - in tyre sizes 26x1.375 is a 559 mm mountain bike tyre but 26x1-3/8 is a 590 mm road bike tyre, the way it's written matters. As it happens there is no 27x1.decimal size so it's not easily confused with a different size but it's still incorrect, used by people or systems that can't enter 1-3/8" or a similar representation of that measurement.]