Did you rule out the most likely causes? Start with these.

Does pop-up drain open enough. If not, adjust.
Check for hair in the tailpiece (just below the drain).
Check for clog in the trap.
Check for clog further down the line.

BTW, how long does it take for it to back up? The shorter the time, the closer the obstruction.

Dump some baking soda in the sink, then, pour 8-10 ounces of white vinegar in the sink. It will drain, but, don't use the sink for an hour or two. If there is blockage, usually this home made remedy will clear it out.

" If there is blockage, usually this home made remedy will clear it out."

only if there is no hoar.

There is no pressure generated, ad the only thing with any real cleaning ability is the vinegar (acetic acid) before it is neutralized making CO2 and foam.

That is not the problem the stopper or plug does not go up high enough is the problem.

If that's the case, adjust it underneath (where the rod connects to the pop-up lever).

It's probably a coincidence, the slow drain thing, unless whoever installed the tap put some material down the drain.

I'd try one of these in the drain: (apologies for the Canadian link, I couldn't find it on homedepot.com)

Here is a link that might be useful: Moen drain cleaning tool

After bathroom remodel and new vanity sink & faucet installed, I have a slow drain - but mostly for warm water only?

There isn't a clog.
The push down/pop-up appears to be sitting high enough.

Cold water drains easily and does not back up past the pop-up, however, Warm water does not drain easily and does back up over top of the pop-up -- very strange.

Apparently I can only post one photo at a time. Above is the warm water issue, showing the back of water. This photo shows that there isn't a problem when only cold water is run.