For a LvL 10 character Base Fighter class lvl 7 (Eldritch Knight), Paladin lvl 2 (for Divine Smite), warlock lvl 1 ( Spell slot regen after short rest)

If you can use War Magic from the Eldritch Knight is a Eldritch blast (warlock) followed with a melee weapon with Divine Smite possible?

IE

Eldritch Blast (2d10) + Rapier (1d8) + Divine Smite (2d8) on a single turn?

Yes, all of that is possible.

Sweet and thanks for the reply, any thoughts on if that can be repeated on an extra attack?

If you use Eldritch Blast you're using the [Tooltip Not Found] action - not the Attack action. Therefore you won't get any extra attacks.

You also only get one Bonus Action each round so you can only make the War Magic attack once.

Edit:
I also want to double check that when you say:

Quote from Ronnie_Soach >>

warlock lvl 1 ( Spell slot regen after short rest)

You do understand that only the Warlock's Pact Magic spell slots will regenerate on a short rest - not all spell slots from the Fighter and Paladin too?

Quote from Emmber >>

If you use Eldritch Blast you're using the Cast a Spell action - not the Attack action. Therefore you won't get any extra attacks.

You also only get one Bonus Action each round so you can only make the War Magic attack once.

Edit:
I also want to double check that when you say:

Quote from Ronnie_Soach >>

warlock lvl 1 ( Spell slot regen after short rest)

You do understand that only the Warlock's Pact Magic spell slots will regenerate on a short rest - not all spell slots from the Fighter and Paladin too?

Yeah, I found in the PHB Errata 
Divine Smite (p. 85). “Paladin spell slot” has been changed to “spell slot.” The following text has been appended to the last sentence: “, to a maximum of 6d8.”

which should allow me to use a Pact Magic Slot, unless I'm mistaken hence the questions

Oh - yea - you can use any spell slot to use the Divine Smite - but only the Warlock Pact Magic ones will come back with a short rest - and for only 1 level in Warlock they'll be level 1 spell slots.

If you are using the character sheet on DDB, spellcasting and pact magic slots should be separate and labeled.

I also want to note, if you take more warlock levels for pact of blade and eldritch smite. Eldritch smite and divine smite can be stacked onto the same attack, but eldritch smite specifically requires the use of pact magic slots.

True True, haven't thought past this point yet. This is thought experiment basically and haven't even thought which Patron the warlock would use much less adding levels to any of the three classes LoL