No, you cannot. Security prevents you from accessing most lounges. The rule is simple: you can access a departure lounge only if you are departing from that terminal. LHR-TLV departs from T4 so that is the only lounge that you can access (you can use the F lounge at T4, which is in the 1st level of the lounge pavilion, below the Concorde Room). You can use the showers in the MB Spa next door to the F lounge but you cannot get a massage (must be flying CW or F on BA for that). Upon arrival you cannot access any of the departure lounges unless you go airside (coming back out will be a major hassle with the current security status so I don't recommend it). AA does have a good arrivals lounge, outside customs in T3, that you can use based on your J boarding card. Good showers, 2 computers (slow) and decent continental breakfast. No massages or hot breakfast like BA offers in its arrivals lounge, but good none the less. Take the HEX over to T4.

The alternative if staying all day at the airport (and that may be a good idea now) is to stay airside, take the bus from T3 to T4 and just use the BA departure lounge at T4. This saves clearing immigration/customs into the UK in order to reach the AA arrivals lounge. My own choice would be to use the arrivals lounge but it isn't compelling.

So you have zero choice in lounges, at least if the rules are enforced (and they have been the past few months). Luckily the lounges you do happen to have access to are the best ones at LHR so it isn't a hardship. But they may be more crowded than normal, with longer stays at the airport, given the circumstances.