The Mallory Court looks very good but although I used to work in Royal Leamington Spa, along with our station cat, a black one called Centro, I don't know it. It is south of town and the easiest (ha) way to the NEC by car is my old commuting route of the A452 through town to the A46 then north to A45 and all the way until the airport/station/NEC.
Only little problems are traffic through Leamington and Warwick, the sneaky speed camera vans on the bridges over the A46, the back ups (and confusing junction - do it the first time each way in the daylight) at the junction of the A46/A45 on the outskirts of Coventry, the speed cameras on the A45, the perennial traffic between Kenilworth Road and Costco/Sainsburys on the A45, and the sky high parking charges at the NEC. (local knowledge and years of experience speaking here)
Counterintuitively, it will be no slower to take the long way around making your way toward Warwick and then following signs to the M40 at junction 14, then M40 to the M42 North confusing junction at M40/M42/M42 because the M40 turns into the M42 towards Droitwich - you want M42 towards Solihull and Airport). Little problems are the high level of traffic on the M42, and variable speed limits controlled by the overhead gantries and associated speed cameras, narrow lanes on the M42 because of the bridge and tunnel work on the boondoggle called HS2 until (or if ever) the construction is finished, backups at the exit for the A45 and the airport/station/NEC, and the sky high parking charges at the NEC.
If you will be there for BBC Gardeners' World Live, it will be packed and parking even harder to get and even more expensive.
A modest proposal would be to park at Leamington station (not free) and take either the local Leamington - Coventry - Nuneaton train to Coventry and change for anything heading towards Birmingham - they will all stop at Birmingham International for the airport and NEC. The airport is reached from the station by a 60 second free ride surface cable car every 90 seconds. The NEC is reached from the station by a fairish walk (depending on the exhibition Hall needed) entirely indoors with some travelators and escalators. Or if you are lucky and get a Cross Country train from Leamington via Coventry (don't take a Cross Country via Solihull, it won't go where you want) because it will whisk you to Birmingham International in only 2 stops, via Coventry but you stay on. Issues on that choice, you need to fit the frequent schedule.