>>2920280
>What kind of glue is best for this?

pic related

The brand is not important as long as it is contact cement

>real wood underneath, but the strip coming off is plastic or vinyl

Contact cement is used to apply laminate to wood countertops - this is what it's made for

>if it needs a long time to cure, how could I apply horizontal pressure on it?

keep the parts separated from each other and apply cement to each side.
wait a few minutes until each side is completely dry
press the parts together and they bond instantly