Using $ct$ axis instead of $t$ axis in special relativity I've recently started studying the concept of space-time diagrams in special relativity, and I came across the concept of representing the time axis using $ct$, with units being that of length. Now I'm told that this is done, first of all, to keep the speed of light = 1, and give a common unit to both the axis, so that we can show, that space and time are inherently the same thing.
However, I'm still not being able to understand, how can I intuitively think of time in meters or any unit of length. Also, doesn't plugging $ct$ and $ct'$ into Lorentz transformations, make it dimensionally inconsistent.
For example :      $x' = \gamma(x-vt)$
if $x$ and $x'$ are in terms of distance, and so is $t$, then the term inside the bracket becomes dimensionally inconsistent.
So, how can I intuitively measure time in meters, or solve Lorentz transformations this way? What exactly does setting the speed of light $c = 1$ mean?
It would be really helpful, if someone can explain to me the motivation, and the intuition behind expressing both axis in the same units. Moreover, what would have happened, if we had kept them in separate units?
 A: This question mixes a few different things.  One is the idea of "natural units" where $c=1$ and another is the idea of scaling the time coordinate $t$ by the speed of light.  These are completely independent.
First, notice that in units where $c=1$ you have $ct=t$.  There's no magic there.  Just substitution.  So the question of natural units makes the scaling $ct$ trivial.  The opposite is not true though.  Making a graph with axis $ct$ instead of $t$ does not imply $c=1$.
Second, there's the question of making the spacetime diagrams that you raise, which could be extended to other conveniences in how to write quantities in special relativity.  In any common convention for units, the quantity $ct$ has units equal to the units for measuring spatial distances. (This is "length" in a general sense, but not necessarily in units you associate with length.  For example, in black hole spacetimes in general relativity, it's common to measure temporal and spatial distances in "units" of the black hole's mass, which requires units with $G=c=1$.)
If you're working in Cartesian coordinates on the spatial dimensions, then your other coordinates $x$, $y$, and $z$ also have units of length.  So for some circumstances, using $ct$ instead of $t$ is handy because it emphasizes that space and time are different pats of a coherent spacetime.  If you're really looking at relativistic effects, it also provides a nice scaling since, e.g., light cones run at 45 degree angles in the diagram while they would be nearly vertical or horizontal, depending on which axis you draw time on, if you used $x$ and $t$ in units like meters and seconds.  But everything in these two paragraphs applies even in units where $c \neq 1$.
Third there's the question of how to make sense of the dimensionless nature of velocity in natural units.  This feels strange when you're new to relativity, but it's actually something that in other contexts is probably quite familiar.  The radian has the same property.  It is recognized as a "unit" but it is also "dimensionless", e.g. https://physics.nist.gov/cuu/Units/SIdiagram.html.  In natural units, speed has this same sort of structure.
A: 
how can I intuitively think of time in meters or any unit of length

A meter of time is the amount of time that it takes light to travel 1 meter. A foot of time is the amount of time that it takes light to travel 1 foot. A light year of time is the amount of time that it takes light to travel 1 light year.

doesn't plugging ct and ct' into lorentz transformations, make it dimensionally inconsistent

No, although you do have to be careful. You cannot just randomly multiply by factors of $c$, you have to multiply by factors of $c/c$. So if you multiply $t$ by $c$ then you have to divide something else by $c$, typically that is velocity so that your velocity becomes a fraction of $c$.
Alternatively, if you are measuring time in units of distance then $c=1$ and furthermore $c$ is dimensionless. So in that case you can throw in factors of $c$ as desired since it is just a dimensionless constant 1. Note, this also means that all velocities are dimensionless and are given as dimensionless fractions of $c$.
