If Chelsea and Arsenal finish outside the top 4 but win the champions league and Europa league respectively. Will England have 6 teams in next years champions league?

No. 4th place team would lose out.

Ok cheers! Highly unlikely but Arsenal fans would enjoy that if Spurs finished 4th.

If only one of Chelsea/Arsenal win their competition and end outside the top 4, then all 5 will be in Champions League next season.

4th place will lose out only if both Chelsea and Arsenal win this season's competitions, as competition rules say 5 teams from any association is the max allowed.