I make lots of use of cable ties, if you still have the trim piece offer it up and see if some suitable holes and half a dozen cable ties would do the job.

Often enough cable ties are more secure than the original fittings which can take no punishment where cable ties are suprisingly rugged, such parts as this under valance are deliberately designed to break away easily if caught so don't cable tie it too well :-)

if secured too well and you caught a kerb for example you could rip the whole bumper off when reversing...anecdote time, Alfa Mito is favourite for this, dead easy to rip the entire bumper off going backwards, seen it done (by others thankfully) several times when reversing off car transporters.

Ebay is your friend, you can buy hundreds of the things really cheaply, i have various sizes and lengths, useful for many jobs about the home cars and garden.

The part probably isn't desperately important, if you lie at the front and wiggle the bumper and trims still there it should be obvious if the trim piece is helping support the others.

My question is - is this part critical and should it be fixed back on immediately? Or can I wait until the next service is due (in a few months time) ?

I believe the part you describe is known as the sump guard. That is one of its functions - another is to smooth air flow under the vehicle and reduce drag. So I would not describe it as 'critical' but it does have a purpose. It also gets in the way if you want to drain your oil properly rather than using suction.

Something similar was ripped off of my VW Up a few years ago. It served to protect the underside of the engine. It's not critical but it is best in place otherwise a branch of other object could end up striking engine components and causing damage. Don't let the garage do it, they'll charge you for a new one, and you'll pay an arm and a leg, along with a few major organs.