In the UK (other countries may vary):
All an employee's rights regarding redundancy etc continue to be the same while furloughed under the Corona Virus Job Retention Scheme:
Employees that have been furloughed have the same rights as they did previously. That includes Statutory Sick Pay entitlement, maternity rights, other parental rights, rights against unfair dismissal and to redundancy payments.
That doesn't mean you automatically come off furlough if you're being made redundant while you are on it - but your redundancy payment would AFAIK be worked out based on your 100% wages not the 80% furlough payment.
Regarding redundancy upon/after the end of the scheme the same page only has the following to say:
When the government ends the scheme, you must make a decision, depending on your circumstances, as to whether employees can return to their duties. If not, it may be necessary to consider termination of employment (redundancy).
There's nothing to indicate that your rights regarding redundancy change specifically at that point - suggesting that the normal terms would apply (i.e. notice period will be either your contractual notice or the statutory period, whichever is longer)
Is this the same if you hand your notice whilst on Furlough? i.e having agreed to take 80% for no work, and then hand your notice in, would you go back to 100% for the period of your notice (again, 3 months in this case).
That's going to be entirely up to your employer, they can take you off-furlough (assuming the three week minimum has been met presumably) but aren't obliged to. If the furlough ends before your notice period does you'll come back and serve the rest of your notice period, except by mutual agreement as usual.