looking at ur symptoms, here's afew thoughts:
Enrichment --------------------------------------------------------------------------
Cold crank enrichment vs running cold enrichment - coolant temp
you mention it push starts fine vs crank starting.
when the starter is engaged, the ECU is told to run a crank enrichment table which might add extra fuel depending on detected temperature.
if the cold fuel trim of a running engine was running way too rich close to flooding, the addition of too much crank enrichment from a faulty coolant sensor might be enough to flood it.
Idle speed too low --------------------------------------------------------------------
Fast idle waxstat: on pre-face engines there's a thermostatic coolant driven fast idle mechanism originally meant to open the throttle slightly below 82c to fast idle but the waxstat tends to seize, stuck in fast idle so most ppl end up closing the idle bleed screw as a bandage fix to slow it down.
The cold fast idle is also done by the IACV. Remedy for a stuck fast idle is to remove the waxstat mechanism to stop it keeping the butterfly open and let the IACV do the same job.
The facelifts don't have waxstats so ignore this.
https://www.micra.org.uk/threads/pollymobiles-rebuild.35251/page-19#post-446256
IACV: the idle air control valve controls the fast idle electronically.
normally when the ECU detects the coolant sensor is cold and is cranking, it fully opens the IACV to raise the idle speed. after cranking and running, the iacv would remain open till the temps reach 82c then closes.
if the iacv is sluggish or seized or miscalibrated, then it won't reach fast idle fast enough or even at all during cranking so there's no enough air to get the engine running. a seized iacv causes the engine to cold idle slowly then rises up once warm. remove and flush clean the valve with brake/carb cleaner, optionally calibrate it.
https://www.micra.org.uk/threads/pollymobiles-rebuild.35251/page-19#post-446256
Combustion Efficiency --------------------------------------------------------------------------
Timing: coilpack facelifts won't apply here. Matching the ecu timing to the crank timing via the dizzy on a preface is important to ensure it fires efficiently and also effects idle speed as a result. when the dizzy timing is wayy off, it might not fire efficiently enough during cold start to get the engine going.
Dizzy cap: coilpack facelifts won't apply here. a strong spark is essential for good combustion, especially during cold starts. the 4 contact points in the dizzy cap tends to build up oxide which increases resistance and weakens the spark, making cold starts harder. so remove and wirebrush the oxide off the points.
MPG issue -----------------------------------------------------------------------------------------------
Closed loop O2: the lamda sensor should heatup and get the ecu to closed loop asap to get it running at a economic 14.7afr
if the O2 doesn't heat up fast enough, the ecu ends up running a rich open-loop more often which wastes fuel.
if the O2 is sluggish or faulty, the ecu fuel trim and AFR will be wayy off and again wastes fuel.
Brakes: check for binding brakes
Tyres: check they're inflated
Alignment: check the wheel toe in/out isn't excessive which could cause alot of scrubbing which increases rolling resistance, wasting fuel and tyres