Unfortunately I think you'll be very lucky to land at 10pm and get to
Blackheath by 11pm.

One thing to take into account is that Heathrow is a nightmare airport
to exit. It can take quite a while to talk from your arrival gate to
passport control. If you have a European Union passport, you can
usually get through passport control fairly quickly. However if you
don't, the queues can often be quite long.

Best case, with an EU passport and hand luggage only, I think you could
possibly be out of customs by 10.15pm, assuming your plane lands on
time.

As outlined elsewhere in this thread, train/tube options would take at
least an hour.

I've found that the best possible taxi time from Heathrow to Blackheath
across central London is about 45 minutes. However this only works in
the early hours of the morning (6am). At 10-11pm on a Friday night,
central London will be very busy and I think it would take about 90
minutes but this is very variable.

Going around the M25 would be quicker at this time of night (though
more expensive). I would guess a journey time of 65-70 minutes (make
sure the driver leaves the M25 at junction 2 - longer but quicker as
compared to exitting at junction 3).

Cheaper than a black cab taxi would be to pre-book a minicab. A company
that provides a reasonable service and is fairly reliable is Blackheath
Cars. I haven't done Heathrow to Blackheath for a while but if memory
serves, the fixed cost (ie there's no meter) is about £55-60 across
central London (more expensive around the M25).

Blackheath Cars can have somebody waiting for you after you exit
customs. Note that you'll lose a few minutes over getting a black cab
as you'll need to walk to where the driver has parked in the short-term
car park and the driver will need to pay for the parking charges.

Blackheath Cars can be contacted on +44-20-8318-5432.

Given the above, on a Friday night, I reckon train and tube options are
better (and cheaper).

Are you committed to Heathrow though? Much more convenient for
Blackheath is London City airport (to which you can fly from several
European locations). You can usually exit City airport in about 5
minutes and a taxi to Blackheath should only take 10-15 minutes
(assuming the Blackwall tunnel isn't closed for maintenance - it does
happen, search the web).

I usually just get a black cab from City airport to Blackheath as the
premium isn't too much over a minicab. Obviously you could also book
Blackheath Cars as above.

Good luck!