Officially the strike ends at 2359 on 8 April, but it would be no great surprise if early services on 9 April were affected.
These cases are always really hard as unlike on National Rail LUL strikes have a habit of being called off at the last moment. Not always, but quite often.
National Express pick up at Earl's Court at 0400, 0505, 0605, 0630 and half hourly for the non stop journey to Heathrow. Depending on the individual journey that is between £6.60 and £10.50 in fare. The buses have started at Victoria.
That is what I would do personally is to remain at Earls Court and take the bus but YMMV.
For a higher £12 fare the bus ticket can be cancelled until 24 hours beforehand, so I would have nothing to lose by doing that.