Is it likely the MAN > LHR flight will be cancelled?
No one knows. LHR has announced that beginning July 11 (and ending September 11, 2022), fewer flights will be permitted, but it'll be up to individual airlines as to which flights will fly and which will not. What British Airways will actually do is up to British Airways. You can find more info on this British Airways page. Were I in your spot, I'd check regularly to see if BA has delayed or cancelled either of your flights. Bottom line: This is a matter of opinion.
It is likely — here's another opinion — that train travel will be less impacted than air travel. You might consider taking the train to Heathrow instead of flying.
Safer to retrieve bags at LHR than check them through to Tel Aviv?
Retrieving and redropping adds several handling steps for the airline's and the airport's baggage handlers. Each step increases the possibility of physical damage as well as misdirection or falling out of a truck or off a conveyor belt. Retrieving and redropping also adds to your duties (standing in lines, carrying luggage from one place to another, dealing with airline or airport staff) during the time you're at LHR, which will likely be crowded and uncomfortable. On the other hand, if BA cancels the flight to Tel Aviv at the last moment, you'd at least have your luggage with you.
Checked-through baggage is much easier for the passenger...if the flight goes as scheduled and the baggage doesn't go astray. While the vast majority of bags make it to their destinations correctly and on time, some don't.
Whether it's safer to check or carry depends upon your risk tolerance and your energy. What's the best for you may well be different from what's best for others.