Fascinating. That spur out of use as a through route by 1900 - and finally closed with the engine shed there in 1966. The WCML▸ bridge, a quad track steel overbridge which has gone.
I assume "spur" means the
LNWR▸ Loop Line, the southern chord, which was a goods line like the
LM▸ Avoiding Line to the north. Both went our of use ages ago, with the Loop Line being just a siding by 1900. The avoiding Line was reopened and re-routed in 2004 via the two new platforms so that
XC▸ can call there. If the bridge over the Loop Line was in need of much work, I can't see how anyone could have justified spending money - after all the Coventry line was closed for a time.
But what's being promoted now is a "dive-under". I can't see anything saying where it is to go, only that burrowing under the main line is going to cost lots. Presumably it would take trains that reverse in platform 1 under the main line and towards Leicester. I suspect better value (and thus probability of funding) would come from doing something different, using what's there already. For example, building (or re-building) a link between other lines to allow a route from Coventry to enter Nuneaton as if from Birmingham.
Or, given a new station was built on the Coventry Line (Bermuda Park), why not build another one one the way to Birmingham? Trains from Coventry could reverse there, which granted would take longer (and might involve two calls at Nuneaton), but you'd be offering the residents of Whittleford a station again.
Note the difference between old and new avoiding lines. The old ones were for goods, and linked lines that would otherwise call for a reversal and/or crossing the main line. Grade separation was secondary. The new ones are all about grade separation, and allowing station calls with a reversal is not a significant penalty. The lines linked may already be linked across a flat junction - which used to be adequate, but now is not operationally acceptable.