LUL & 25kV Electrification

Well, in the tubes I guess there would be the question of whether there is room for it.

I know that overhead rails were considered as part of the Victoria line "Space Train" proposal.

My question is more aimed at (for example) whether overhead electrification of the Chiltern lines would be workable, or would this lead to issues with Metropolitan trains (still using live rails)?

Quite straightforward: there are two approaches: double element relays (DEV), or filtered feeds.

DEV relays use a control and a local core to remove extraneous induced current. The local supply must be screened from the 50Hz, but the phase angle between the two supplies must be at 90° for the relay to pick up (left hand rule), DEV relays must have matching frequencies in both cores and at right angles to give a positive torque to the vane.

Filtered feeds are a lot more of a faff, with oscillators and amplifiers at the feed end. You need to avoid self-oscillating components, solid-state failures must cause the relay to drop (so the supply needs to be AC).

There is a lot more to it, but it is technically straightforward.

Apart from the signalling issues already mentioned the big issue with overheads is what happens to NR fault currents directed to earth at or adjacent to LU lines.
Just as in a domestic electrical supply situation, there are two approaches to protection, protective earth bonding and isolation.
In the case of King's Cross to Liverpool Street everything is bonded and as already mentioned coded track circuits are used but other things worth pointing out are the fact that working on equipment can be hazardous in such areas and the earth bonds must be maintained when changing equipment, generally where this applies traffolyte labels with suitable info are attached to the affected equipment. Also the main telephone cable in that area is of the armoured variety to give additional electrical and mechanical protection in the event of large earth fault currents flowing.
The District line similarly has armoured telephone cables and indeed the main cable through Campbell Road is/was an armoured submarine cable. I recall having to divert that when I was diverting comms cables during the Campbell road bridge replacement a few years ago now.
HEX (Heathrow Express) affects quite a portion of LUL including Ealing Broadway to Hanger Lane and also at Kensington Olympia IIRC as well as Heathrow though I am not aware of any specific special measures as such apart from the tunnel telephones which use bunched 8 pair/0.9 cables instead of open tunnel wires (same system used between King's Cross & Liverpool Street) and of course all the signalling uses double cut and fault screen circuitry similar to that on the Victoria line, I guess the planners were expecting HEX long before it arrived!
In the Ealing Broadway area Ealing Broadway itself is all bonded out to a spider plate and NR earth fault currents can be measured locally on equipment provided there. An isolation gap was provided beneath Hanger Lane junction on the Central line such that there are no earth connections between the District and Central line at that point. The sheaths of all the cables at this point had a section removed to ensure that cables could not carry stray earth currents between lines.
When rerailing between Ealing Broadway and Hanger Lane junction no rails should be removed without first applying temporary bonding to maintain the protective earth and all track circuits are double bonded both rails with red bonds and labelled. The temporary protective earth bonds (about 100' long IIRC) have to be hammered on and hammered off the rails are they are a sprung clip arrangement. No conductive items such as piles of rails are supposed to be left in the area unless they are bonded to the protective earthing system!

A poorly written paragraph on my part, there was no suggestion of double cut circuitry anywhere except at Heathrow, however, rereading what I penned rather hurriedly does allude to that impression!

While I know that double cut was in vogue at the time that Heathrow was built and the fault screen was probably more to do with concrete tunnel linings than a future HEX I believe there was more to it than that. The Picc only ran as far as Hatton Cross when I started with LT in 1977, indeed there was a signals installation school there back in 1978 to which many new starters were sent to learn equipment wiring although I never had the pleasure as my 'training' was 'on the job' on the Central line resignalling which was reaching a conclusion and the Northern line resignalling which was ongoing and on stage 1 at Cockfosters and Oakwood but the west end of the Picc still had some ongoing work at Northfields and Boston Manor as well as Heathrow.
My thoughts were more about the T4 loop, T4 opening in 1985 IIRC and HEX in 1988. The tunnel telephone system was the point here, having no open T/T wires exactly as on the H&C adjacent to the CWL and also installed in the 1980s.
In the light of the work that had to be done at Ealing Broadway and Hanger Lane etc had the Picc at Heathrow been signalled and equipped using the older signalling standards I have always thought that work would've been required to mitigate any interference from HEX and clearly I made a hash of saying that.

I apologise for any confusion.

Yes, I know that!

Surely it was a bit short sighhted to remove the third rail from that part?

There are a number of issues causes on both sides with track circuits dc stray currents can effect track circuits in ac electrified areas and vis versa.

One of the main issues regarding 25kV as is the traction return, at the return system is basically earthy as all the structures are buried direct into the earth all running rails are bonded to traction return either one is directly or as in the HEX area via impedance bonds bridges etc are also bonded to traction return.
When two electrification systems come in to close proximity and that does not matter if its dc / ac or ac / ac North Pole Depot is segregated electrically from the GWML by isolation transformers.

When NR dc and ac systems meet often a system of isolation transformers is used there lots of these around the NNL and WLL to segregate track circuit issues, at Blackfriers for Thameslink a system of sequential contactors and rectifiers is being used.

Where NR and LU come into close proximity the aim is separation to avoid touch potentials by distance greater than 2 meters. When like at Ealing and West Hampstead where the LU crosses LU and the separation can not be achieved a "Faraday Cage" is used the bridge in the above cases are bonded to the LU system in their normal way and the Faraday cage is bonded to NR system to prevent flash overs developing spark gaps are installed that trigger at about 3000v

The whole ac dc interfaces is a very complex and specialised area the testing for HEX took months at Ealing to get the correct data to satisfy the HMRI