I would suggest to fly BD to London, but join LH program. This is very good for Israelis, as it treats TLV as in Europe. So flysmart awards to Europe are only 15K in Y, and business class to US is only 90 K with the companion 50% off (and kids 25% off). You would get to SEN pretty quickly. Soon you will also have the option of earning/redeeming on CO - you can already earn/redeem on AC, US, LH, LX, OS, BD, TK etc to USA.
Soon you may even get HON points on BMI, in which case HON may even be within reach - if you have a chance, go for it. I know of commuters to London who choose to fly via FRA because of the service (and the HON D class seat guarantee etc).

BD program is excellent, but awards to/from Israel are quite expensive, and who knows how long it will last. There are also a lot more BD fares which earn very few miles.

If you do fly LH, look at joining their PPB program also (www.ppb.co.il). It's free, and the points are very useful for upgrades to/from TLV.