This is a very difficult question to answer and I doubt if anyone actually knows and I guess we're talking wild deer and not park situations.
There are several areas of England where there are at least five species present but it would need fallow or reds to be included to get the numbers up. There is one estate in Sussex where fallow predominate but there are also shootable numbers of roe, you would see muntjac on most visits, there are a couple of sika hinds running with the fallow and the odd sika stag comes in for the rut plus there is the odd red in the general area. I've seen all bar a red in one morning's stalk there and the professional stalker had a client out one day when I was also there, he was stalking a fallow buck, whistled to stop the intended target and a roebuck and sika stag both stood up.
In East Anglia, there are five of the six UK species present (no sika as far as I know) but what the density is and whether all five occur in the same location I don't know, there aren't too many fallow so densities may not be that high.
There are five of the species in the New Forest (no CWD) and given that there is a very large density of fallow, plus red, sika, roe, and muntjac my guess is that the area with the most deer of different species would be somewhere in the New Forest or immediate environs.
Other candidates would the somewhere in the Marches or Cotswolds where there can be very high fallow densities plus some good roe and muntjac numbers and I know an area where there are also some reds and sika. Areas of Dorset will have roe, fallow, sika and some muntjac, some parts of Devon have high densities of reds, especially if there is a LACS sanctuary in the area, and there will often be roe as well, in Devon the sika and fallow are not normally close to the reds. A lot depends on the level of culling going on in any particular area.
There is hill stalking for reds in the Lake District.