To ensure the lip sync setting doesn't need to be altered between normal TV sound output and headphone use, I recommend you connect any headphones to your TV not the Humax (this has the additional benefit of being able to use the same arrangement with another source, eg live TV from the TV's own tuner, or a DVD player).
Connecting to the TV will be more convenient anyway, because TVs often have a 3.5mm stereo output for that purpose, while the Humax doesn't and you would need to adapt from the analogue audio signals on the SCART (if it has one) or phono connectors. I would go so far as to say that, in my experience, all wireless headphones have a 3.5mm stereo jack to connect the input, as do wired headphones these days (¼" jacks would have been common in the past, but few items of audio kit outside the professional arena still use them).
If you did decide to connect to the Humax, a wireless set would probably work (with a suitable adapter), but wired headphones would need some kind of amplifier and volume control.
There is no problem obtaining wireless headphones, they are even available from Lidl and Aldi from time to time. It's more a question of how much you want to pay. As the headphones run from batteries, look at what type they are. Headphones with built-in non-replaceable batteries (typically charging from the stand or a USB adapter) would be difficult to use while the batteries are charging, and the service life is limited by the life of the battery. Although at first glance less convenient, headphones that use standard AA or AAA cells can use just about any battery, and using NiMH and external charging means you can have one set in the headphones and another set charging, and replace them when they fail. And, if desperate, can always fit standard alkaline batteries in an emergency.
Then there's analogue v digital. Analogue transmits on a licence-free frequency using FM to (essentially) an FM receiver in the headphones. Sometimes the headphones can also be used for broadcast radio. Digital uses (typically) Bluetooth. I recently acquired some Bluetooth headphones being sold off at Tesco, and (separately) an audio to Bluetooth adapter. With the headphones paired to the adapter, I can plug it into any audio source and power it from a USB socket (modern TVs have USB sockets) and just pick up the sound in the headphones... or play MP3s from my phone just as easily. (The Bluetooth adapter works the other way as well - able to pair to a sound source and inject audio into an amplifier and speakers for example).
The downside of digital sound is that it may be accompanied by a lag. This is irrelevant for sound only, but matters when the sound has to be in sync with vision. I haven't noticed a problem with my Bluetooth headphones (but haven't used them much), and note that sound after vision is natural and much more acceptable than sound before vision.