Actually.............. with a new kitchen being installed, there is no problem.
Tile the whole floor before installing the kitchen. Then, there are no height issues.
Even if you decide to tile after the kitchen is in, it's simple. All kitchens have adjustable legs. You just have to get the fitter to jack up all the units to a height where appliances fit under the worktops normally.
I don't like to be awkward and go against what the others have said, but, as for the threshold to the living room, I'm with Arrods.
A standard 150mm by 20mm thick hardwood threshold (available off the shelf) is what I've used several times. You just need to shorten the door.
It's purely your choice, Needy. Removing the old tiles is the pukka option, but I just wanted to show that there are alternatives.