The following advice from my sister, who used to work for a magazine which specialised in grass*...
The grass should bounce back a bit, once it has dried out. although it is tempting to cut it now, I'd seriously recommend that you leave cutting it until there has been a sustained period of dry weather. This will allow the grass to rise up slightly, which will make the cut more effective.
You can also get a set of spike for the front of many mowers which 'lift' long grass that has been flattened. Alternatively, rake it over with a flexible rake before cutting.
After the first cut, leave it for 3-4 weeks, watering if needed, and then cut it again. After the second cut, feed it with a general purpose lawn feed.
*Yep. There are actually magazines that specialise in such things. My sister was inordinately proud when her publication appeared in the 'Missing Words' round on Have I Got News for You...